The possibility of Mr Romney being offered a government job or task force was raised by the president in his victory speech on Tuesday night.
"I also look forward to sitting down with Governor Romney to talk about where we can work together to move this country forward," Mr Obama said.
He told them: "Thank you my friends, thank you so very much. I have just called President Obama to congratulate him on his victory.
"I wish all of them well, but particularly the president, the first lady and their daughters.
"This is a time of great challenges for America, and I pray that the president will be successful in guiding our nation."
Mr Romney thanked his running mate Paul Ryan, saying he hoped he would continue to use "his intellect and his hard work and his commitment to principle" for the good of the nation.
After paying tribute to his wife Ann and his family for their support, he went on to thank his campaign team for their "extraordinary effort, not just for me but for the country that we love".
Meanwhile, Grant Bennett, who served alongside Mr Romney as a Mormon bishop in Massachusetts, said his friend would not run for office again but could be persuaded back into public service.
"If Mitt were invited by the president, he would say 'I'm an American. Can I contribute?'," Mr Bennett said.
"Mitt's fundamental motivation is to make a difference for good."
Such a move would be likely to anger Republican colleagues determined to block Mr Obama from implementing his plans. Alex Castellanos, a prominent Republican strategist, described Mr Obama's gesture to Mr Romney as "incredibly generous, but also incredibly smart politics".
Mr Romney is expected initially to retreat to his beach-side home in California, and spend time with his five sons and 18 grandchildren.
But Mr Bennett said that at 65, Mr Romney "has enormous energy and is in great health" and would not be retiring.
He said Mr Romney, who has a $250m (€196m) fortune, was "not at this stage concerned about money", and could devote himself to charity work rather than the private sector.
Yet a former senior aide to Mr Romney's father, George, struggled to identify a cause with which Mitt Romney might associate himself, especially after dismissing poorer voters as government-dependent "victims" in secretly-recorded remarks during his campaign.
Mr Bennett suggested that Mr Romney may return to a position in the Church of Jesus Christ of Latter-Day Saints, to which he has continued donating millions of dollars. (© Daily Telegraph, London)