Anti-Trump and pro-Trump signs compete outside Trump Tower in New York. Demonstrations have continued in cities across the country in protest of Donald Trump’s rhetoric and personnel choices. (Jewel Samad/Agence France-Presse via Getty Images)

President-elect Donald Trump and Russian President Vladimir Putin agreed in a telephone conversation Monday that relations between their countries were “unsatisfactory” and vowed to work together to improve them, the Kremlin said in a statement.

Moscow said the two men discussed combining efforts in the fight against terrorism, talked about “a settlement for the crisis in Syria” and agreed that their aides would begin working on a face-to-face meeting between them.

Trump’s office later said that Putin had called to “offer his congratulations” and that they had discussed shared threats and challenges, “strategic economic issues” and the long-term relationship between the two nations.

On Tuesday, a Kremlin spokesman said that Putin and Trump had not discussed the Russian annexation of Crimea from Ukraine in 2014 or the rebellion by Russian-backed separatists in eastern Ukraine.

“The overall tone of the conversation corresponded with the tone of statements already made during [Trump’s] election campaign concerning Russian-American relations,” the spokesman, Dmitry Peskov, told journalists.

During his campaign, President-elect Donald Trump repeatedly vowed to "drain the swamp" in D.C. and rid the federal government of political elites and lobbyists. But just days into his transition to president, Trump seems to be doing the opposite. (Deirdra O'Regan/The Washington Post)

During the campaign, Trump said that he “would look at” recognizing Crimea as a part of Russia. The United States and the European Union issued sanctions on Russia over Crimea, and broadened them as a result of Moscow’s involvement in eastern Ukraine.

The president-elect spoke admiringly of Putin during the campaign, praising him as a stronger leader than President Obama and saying the two countries should join together to fight terrorists, particularly the Islamic State in Syria.

Those views put Trump at odds with many GOP defense hawks, who have praised his promise to increase military spending but are uniformly suspicious of Moscow and have denounced Russian actions in Eastern Europe, Ukraine and Syria. The offer of cooperation could also immerse Trump in a deep controversy with the Pentagon, where military and civilian leaders have strongly opposed collaboration with Russia, particularly in Syria.

U.S. intelligence officials have also expressed concern, noting that the Kremlin is believed to have been involved with hacking the email accounts of prominent Democrats, in hopes of injecting chaos into the U.S. electoral process and perhaps swaying the outcome of the vote.

Trump’s conversations with Putin and other world leaders came as protests continued for a sixth straight day in major cities and on college campuses over last week’s election results, in which Trump won the electoral college but lost the popular vote to Democratic nominee Hillary Clinton.

[Trump called these foreign leaders the day after the election]

He also faced escalating criticism over his appointment of former Breitbart News head Stephen K. Bannon as chief strategist at the White House. Bannon has been denounced by a chorus of advocacy groups, commentators and congressional Democrats as a proponent of racist, anti-Semitic and misogynistic views.

“Bringing Steve Bannon into the White House is an alarming signal that President-elect Trump remains committed to the hateful and divisive vision that defined his campaign,” House Democratic Leader Nancy Pelosi (Calif.) said in a statement. “There must be no sugarcoating the reality that a white nationalist has been named chief strategist for the Trump Administration.” Trump allies have dismissed the accusation.

Obama, in his first news conference since the election, declined to comment on Bannon and sought to reassure the country and the international community that Trump is committed to governing in a more realistic and pragmatic fashion than he displayed on the campaign trail. Obama said Trump pledged in their conversation last week to maintain U.S. strategic relationships, including the NATO alliance.

At the same time, Obama urged the president-elect to reach out to groups representing minorities and women, many of whom have felt slighted by his candidacy.

“One of the great things about the United States is that when it comes to world affairs, the president, obviously, is the leader of the executive branch, the commander in chief, the spokesperson for the nation, but the influence and the work that we have is the result not just of the president. It is the result of countless interactions and arrangements and relationships,” Obama said. “. . . And there is enormous continuity beneath the day-to-day news that makes us that indispensable nation when it comes to maintaining order and promoting prosperity around the world. That will continue.”

Trump has signaled in numerous ways that he will not hew to many of the norms of past presidents, from declining to release his tax returns to relying on his family as key advisers. Previously, he has said he would avoid conflicts of interest as president by turning his business over to his children.

[Trump’s win may just be the beginning of a global populist wave]

Since his victory last week, Trump has received congratulatory calls from a number of foreign leaders, including some he sharply criticized during the presidential race, such as German Chancellor Angela Merkel. Sunday night, after he spoke with Chinese President Xi Jinping, Trump’s office said that he “believes the two leaders will have one of the strongest relationships for both countries moving forward.”

Trump’s relationship with Russia may be one of his trickiest to navigate.

Although the statement released by his office contained few details and did not mention Syria or other specific issues, it said Trump told Putin “that he is very much looking forward to having a strong and enduring relationship with Russia and the People of Russia.”

In addition to praising Putin, Trump has indicated that closer relations with Russia would keep the Kremlin from establishing tighter ties with China. He has appeared to absolve Russia from responsibility for intervention in Ukraine and the annexation of Crimea, and during the campaign repeatedly questioned the relevance of NATO, which has charged Moscow with engaging in provocative air and sea actions on the alliance’s eastern flank.

Giving Putin a free pass on those issues would run directly counter to the Russia policy of the Obama administration, which has, among other things, called for an international war-crimes investigation of Russia’s actions in Syria. It could also undermine current European negotiations with Moscow over Ukraine, as well as support for U.S. and European Union sanctions.

Russia is interested not only in getting the sanctions removed but also in obtaining equal status as a player in the Middle East and elsewhere.

Moscow is the primary supporter, along with Iran, of President Bashar al-Assad. In September 2015, one year after the United States began bombing Islamic State targets in Syria, Putin ordered Russian jets into operation against “terrorist” targets there.

Most Russian airstrikes, however, have targeted U.S.-backed opposition forces fighting against Assad and, according to the United Nations, have killed hundreds of civilians. The United States and its European allies have called for an international investigation of Russia for war crimes.

Obama administration offers to coordinate U.S. counterterrorism strikes against the Islamic State in Syria with Russia, if the Russians would stop bombing opposition and civilian targets, have repeatedly failed to stop the carnage. Any cooperation with Russia would also put the United States on the side of Iran, whose activities in the region Trump has vowed to stop.

Also controversial was Trump’s decision Sunday to name Bannon as chief strategist, a move condemned by the Anti-Defamation League, the NAACP, Capitol Hill Democrats and some Republican Trump critics.

Trump’s staff brushed off the criticism Monday, saying Bannon is a former naval officer and Goldman Sachs executive who has been a measured voice behind the scenes at the campaign.

“That’s not the Steve Bannon that I know, and I’ve spent a lot of time with him,” Trump’s new chief of staff, Reince Priebus, said on MSNBC, responding to allegations of racism, misogyny and anti-Semitism. “. . . He was a force for good on the campaign at every level that I saw, all the time.”

Kellyanne Conway, who worked closely with Bannon as Trump’s campaign manager, also defended him. “He’s been the general of this campaign,” she told reporters as she arrived Monday at Trump Tower in Manhattan to meet with the president-elect.

Asked whether Bannon needed to explain his connections to the alt-right movement, which includes many white nationalists and white supremacists, Conway said: “I’m personally offended that you think I would manage a campaign where that would be one of the going philosophies. It was not — 56 million-plus Americans or so saw something else. . . . You should really focus on the will of the people, which was to elect Donald Trump the president.”

House Speaker Paul D. Ryan (R-Wis.) sought to calm the fears many Americans still hold about Trump’s election.

“There is a lot of hysteria and hyperbole,” Ryan said during an interview Monday with his hometown radio station, 1380 Big AM in Janesville, Wis. “I would tell people to just relax — things are going to be fine.

Jerry Markon, Juliet Eilperin, Greg Jaffe, Andrew Roth and David Filipov contributed to this report.