Obama called Merkel to wish her a speedy recovery after her December skiing injury and invited her to visit at a "mutually agreeable time in the coming months," the White House said in a statement.
Merkel last year reacted furiously to claims the National Security Agency had been listening in on her mobile, telling Obama in October that this would be a "breach of trust" between two allies.
Media reports of American snooping based on documents leaked by fugitive intelligence contractor Edward Snowden have damaged US relations with key allies and were a political and personal embarrassment for Obama.
The invitation to Merkel comes as the White House tries to draw a line under the Snowden issue, with Obama poised to give a speech to Americans this month detailing how the NSA's massive phone and data collection activities will be reformed.
"The president spoke to Chancellor Merkel today to wish her a speedy recovery following her injury and to congratulate her on the formation of her new cabinet," a White House statement said.
"The leaders noted the full agenda for 2014, including the Transatlantic Trade and Investment Partnership (T-TIP) negotiations and NATO Summit, and looked forward to working closely together to advance our shared interests."
Merkel's office confirmed in a statement that she would accept Obama's invitation to visit the United States, but like the White House, did not offer a date.
The coming meeting may be a sign that neither side wants the NSA issue to disrupt what may be the most important relationship between the United States and a nation on continental Europe.
"This is the best way to try to begin to mend fences over the NSA affair," said Stephen Szabo of the German Marshall Fund of the United States.
Szabo said the rapprochement stood in sharp contrast between the freeze that enveloped US-German relations after former president George W. Bush and former Chancellor Gerhard Schröder clashed over Iraq.
The White House would not say whether Obama and Merkel had discussed the NSA phone tapping allegations in their telephone call.
One reporter raised a laugh among his peers in the White House briefing room, however, when he asked whether the call had taken place on a mobile phone or a fixed landline.
The discomfort between Obama and Merkel over the NSA allegations is all the more notable because he has privately confided she is one of the foreign leaders whom he most respects.
The president honored Merkel with a White House state dinner in 2011, and she repaid his hospitality by welcoming him to Berlin for a long awaited visit last year, which included an evocative speech by the US leader before the Brandenburg Gate.
The NSA allegations were especially damaging in Germany due to sensitivity over mass state spying on citizens by the Stasi secret police in the former communist East.
They were also a political embarrassment for Merkel, and she demanded answers over what she said were "grave" allegations that tested transatlantic ties ahead of crucial trade talks.
"They must be explained and, more important still for the future, new trust must be built," she told the German parliament.
Some observers and politicians in Washington however accused the German leader of naivety in thinking that US, and other security services were not interested in intercepting her communications.
And senior US officials have signaled that they will not agree to a blanket "no spying" agreement between the two nations.