"We'll give it until tomorrow," Rep. Mike Conaway, the Texas Republican who is running the House panel's Russia investigation, told CNN Thursday.
Asked if the panel would subpoena Stone if he did not disclose the name by Friday, Conaway said: "Yes. We'll take the next steps. Hope we don't have to."
California Rep. Adam Schiff, the committee's top Democrat told CNN: "We have agreed to subpoena him if he doesn't provide the information. ... We'll see whether he's willing to comply or whether we'll have to use a compulsory process."
Grant Smith, a Stone attorney, said: "We are working to comply by the deadline set by the committee. Everything has been collegial and professional. There has been no threat of a subpoena."
Stone, who has known Trump for 40 years, testified before the House intelligence committee in a closed session last month.
Investigators asked questions about his contacts with the Russian hacker Guccifer 2.0 during the election season, something that described as an innocuous and meaningless exchange over Twitter. US intelligence community said the name Guccifer 2.0 was used by Russian intelligence to disseminate stolen emails.
And he insisted he did not collude or coordinate with Russians as part of an effort to meddle in the campaign.
But after the hearing, Stone told reporters that he answered all of the committee's questions but one: his "intermediary" to Assange.
Stone said his conversation with the intermediary was off-the-record and with a journalist, and he would honor the agreement to protect the journalist's identity. But he added that he planned to go back to the intermediary and ask to release him from the off-the-record agreement.
"I'm not going to burn somebody I spoke to off the record," Stone said. "If he releases me, if he allows me to release it, I would be happy to give it to the committee. I'm actually going to try to do that."
Conaway said Thursday that the name "will help eliminate any questions people might have, and it just helps support the story."
"If we've got that name, then we can just evaluate what he said as a result of what he said, I think that will be helpful to the investigation," Conaway said.
In his opening statement for the hearing, Stone denied any direct contact with Assange and attacked Schiff for suggesting he did when former FBI Director James Comey testified in March.
"On June 12, 2016, WikiLeaks' publisher Julian Assange, announced that he was in possession of Clinton DNC emails. I learned this by reading it on Twitter," Stone said.
"I asked a journalist who I knew had interviewed Assange to independently confirm this report, and he subsequently did," Stone wrote. "This journalist assured me that WikiLeaks would release this information in October and continued to assure me of this throughout the balance of August and all of September. This information proved to be correct."
"I have referred publicly to this journalist as an, 'intermediary,' 'go-between' and 'mutual friend.' All of these monikers are equally true," Stone added.
During the presidential campaign, Stone appeared to predict on a few occasions that WikiLeaks would release damaging information about Democratic presidential nominee Hillary Clinton, including saying that it would soon be Clinton campaign chairman John Podesta's "time in the barrel."
Stone denied that he had any advance knowledge that WikiLeaks would release Podesta's emails, however, saying it was a reference to his own investigation into Podesta.