Last week, a mostly Democratic group of House members called on Mr. Trump to rescind his invitation to Mr. Erdogan. On Wednesday, the Senate Republican leader, Mitch McConnell of Kentucky, issued a statement saying that he hoped the meeting would help to restore relations with Turkey. But, he noted, “I share my colleagues’ uneasiness at seeing President Erdogan honored at the White House.”

Although Mr. Trump briefly threatened Turkey with crippling economic sanctions after its Oct. 9 invasion of Syria, he had also cleared the way for the incursion by withdrawing American forces after a phone call with Mr. Erdogan. On Wednesday, he returned to a warmer tone, stressing the value of the decades-long strategic relationship between the two countries — both NATO members, but ones that have drifted apart over the past decade.

During remarks to reporters over the course of a day of diplomacy largely overshadowed by an impeachment hearing on Capitol Hill, Mr. Trump said he hoped that the United States and Turkey could work out their differences. The president said he wanted to increase trade between the countries by a multiple of four, to about $100 billion, “which would be great for Turkey and great for us.”

Mr. Trump also expressed confidence that Washington and Ankara could resolve their standoff over Turkey’s purchase of the Russian missile system, which leading Republicans in Congress call a clear violation of a 2017 law requiring sanctions on countries that buy military hardware from Moscow. Some members of Congress have suggested the sanctions might be avoided if Turkey does not power up the missile system.

Mr. Trump called five Republican senators to the White House on Wednesday, including some sharp critics of Mr. Erdogan, for an unusual meeting with the visiting Turkish leader.