If Ryan were to run, he would face a significant uphill climb against Nancy Pelosi, who has deep support within her caucus and raised millions for her colleagues. Leadership elections are scheduled for Thursday.
"Congressman Ryan is flattered that a growing number of members of the Democratic caucus have called on him to run for leader," Ryan spokesman Michael Zetts said. "He understands that many members are deeply concerned about the future of the Democratic party and caucus. He watched many traditional Democrats leave our party and he is concerned that if changes aren't made we will be in the political wilderness for many years to come."
The statement went on to say Ryan "has the highest respect for Leader Pelosi," and that "he considers her a friend and mentor.
Ryan could also face some questions about his level of support for House Democrats, with one senior Democratic aide noting that the Ohio Democrat only paid half of the $200,000 in dues he owes the Democratic Congressional Campaign Committee this cycle.
In a letter, more than 20 House Democrats -- led by Massachusetts Rep. Seth Moulton -- are urging House Democratic leaders to delay this week's leadership elections, as the party struggles to find a new way forward after its stunning rebuke at the polls last week.
Senior Democratic officials say they expect the leadership elections to take place Thursday as planned.
But the letter indicates some level of concern within party ranks about whether the Democrats' current crop of leaders are the right people to bring the party back into power. Some of the signatories include members elected within the last few election cycles, underscoring a generational divide within the House Democratic Caucus.
Ryan signed onto the letter urging for a delay in the leadership elections. But his spokesman said he pulled his name so as not to create an apparent conflict.
In the letter, Moulton's group told Pelosi that Democrats find themselves in a "difficult situation" and that more time is needed for an "extensive conversation" between the rank-and-file and party leadership. He asked for leadership elections to be delayed until after Thanksgiving.
"It is vital that our Caucus take the time to listen to the American people and learn the lessons of this difficult election in order to put our Caucus in the best position to fight the potentially dangerous agenda of President-elect Donald Trump and to have a realistic chance of taking back the House in 2018," Moulton said.
Moulton added: "We feel very strongly that our Members deserve the opportunity to thoroughly discuss the results of the election and analyze how our party and the Democratic Congressional Campaign Committee's strategy fell so short of our goals."
A Moulton spokesperson said that more than 20 Democrats have signed onto the letter.
Asked for comment, a Pelosi spokesman referred questions to a letter the Democratic leader sent last week saying leadership elections would take place this week.
Pelosi still has significant support from House Democrats, given her fundraising prowess and that her progressive politics are in line with much of her caucus. On Friday, 40 House Democratic women wrote a letter urging Pelosi to stay as Democratic leader next Congress. So there's little chance of anyone unseating Pelosi atop the Democratic leadership.
But Pelosi's long tenure running the House Democratic Caucus has prompted some grumbling among younger members.