(CNN) Senate Republicans blocked two moves by Democrats Monday to try to reverse President Donald Trump's controversial executive order restricting travel to the United States by Muslims and refugees.

In a short exchange on the Senate floor, Senate Democratic Leader Chuck Schumer sought unanimous consent to begin debate on a bill authored by Sen. Dianne Feinstein, D-California, which would rescind the executive order.

Schumer noted that several Republican senators had raised concerns with the order and how it was developed.

"So let's repeal the order and then sit down and discuss a smart, thoughtful, effective way to counter terrorism," the New York senator said.

Speaking for Republicans, Sen. Tom Cotton of Arkansas, objected.

In a second request, Schumer asked that until the travel restriction is lifted and until secretary of state nominee Rex Tillerson explains if he supports the executive order and if he helped in its development, the Senate put off a procedural vote to advance Tillerson's nomination.

Cotton rejected that too.

"It is not a ban," he said. "It is simply a temporary pause for three to four months to evaluate whether Obama administration policies are strong enough to keep this country safe."

House Democrats attempted to get unanimous consent to bring up a similar measure that would defund and repeal the president's executive order today, but their request was denied.

Schumer announced in a Facebook post earlier Monday that he would oppose all nominees who back Trump's executive order.

"I've made it very clear I will vote NO on nominees (Betsy) DeVos (education), Tillerson (state) and (Jeff) Sessions (attorney general)," he wrote. "Nothing will change that, and while I will continue to demand that each nominee issue a public statement on his or her views of President Trump's Muslim Ban, I will vote against nominees who will be the very worst of this anti-immigrant, anti-middle-class, billionaires' club cabinet."

Last week, Tillerson won the backing of the Senate Foreign Relations Committee to be secretary of state when it voted 11-10 along party lines to send his nomination to the full Senate.

The former oil CEO's path to being confirmed as the chief US diplomat hasn't all been smooth sailing.

Florida Sen. Marco Rubio was a long hold-out and potential "no" vote after he raised concerns about Tillerson's views toward Russia and human rights. But the Florida Republican announced the day of the committee vote that he would support the nomination. If Rubio had voted no, it could have stalled Tillerson, although GOP leaders planned to move his nomination to the floor regardless.

Democrats have catalogued a series of concerns about Tillerson, including his stance on Russia and relationship with President Vladimir Putin, human rights, how long he would recuse himself from decisions that could affect ExxonMobil, and the oil company's track record of doing business with despots.