WASHINGTON — One month ago, President Trump warned President Recep Tayyip Erdogan of Turkey not to “be a fool” or “a tough guy” by invading northeastern Syria. The Turkish leader sent his troops across the border anyway, outraging much of official Washington.

On Wednesday, Mr. Trump seemed unbothered by the episode and gave Mr. Erdogan a warm welcome, offering little sign of frustration with Turkey’s authoritarian leader over an incursion that scrambled American policy in the region.

“I’m a big fan of the president,” Mr. Trump said during a joint news conference at the White House with Mr. Erdogan. “My dear friend,” Mr. Erdogan said of Mr. Trump.

Mr. Erdogan’s invasion of Syria, in part to attack Kurdish fighters who helped the United States crush the Islamic State, infuriated many military and diplomatic officials and prompted members of Congress to draw up punitive sanctions. Many of the same people also are angry over Turkey’s recent purchase of an advanced Russian antiaircraft missile system, which legislators from both parties call grounds for separate sanctions.

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.

“I project that we will work something out,” Mr. Trump said. Also in the meeting were Senators Joni Ernst of Iowa, Jim Risch of Idaho, Ted Cruz of Texas, Rick Scott of Florida and Lindsey Graham of South Carolina.

Mr. Graham, for one, betrayed a note of amazement over dropping in on a foreign leader’s White House visit.

“I’ve never had an opportunity like this before,” Mr. Graham told reporters who were briefly ushered into the room. “I appreciate it. The purpose of this meeting is to have an American civics lesson for our friends in Turkey. And there’s a pony in there somewhere if we can find it.”

To which Trump quickly replied: “And I think we will.”

Mr. Trump repeatedly sounded sympathetic to the Turkish leader, seconding Mr. Erdogan’s complaints that European nations have not contributed enough to the cost of caring for more than three million Syrian refugees who have streamed across Turkey’s border since their country plunged into civil war more than eight years ago.

Mr. Trump also sided with Mr. Erdogan after the Turkish leader sharply criticized President Emmanuel Macron of France, who last week called Mr. Erdogan’s invasion of Syria a dangerous mistake that will empower Iran and Russia.

“I think that bothered the president very much,” Mr. Trump told reporters. “I think a lot of other people feel that way, too.”

Mr. Macron also said the North Atlantic Treaty Organization was experiencing “brain death” because of Mr. Trump’s policies.