WASHINGTON – Stung by another health care defeat this week, President Trump said Wednesday he will begin talking with Democrats on legislation that can replace the law signed by predecessor Barack Obama.

"I will negotiate with Democrats to see if we can make a bipartisan bill," Trump told reporters at the White House, the day after Senate Republicans' last-ditch proposal to unwind the Affordable Care Act collapsed on Capitol Hill.

The president said he hopes for a health care vote in January, February or March.

In the meantime, the president said he plans to sign an executive order – likely next week – that would enable people to buy health insurance across state lines, though there is some question as to whether a president has the authority to effect such a change.

"I'll probably be signing a very major executive order where people can go out, cross state lines, do lots of things, and buy their own health care," Trump said.

More:RIP, repeal and replace: Republicans' last-ditch effort on health care is dead

Trump also insisted on Wednesday that Republicans do "have the votes" on health care, but don't have the ability to get it done before a key deadline on Friday.

Since the election, Republicans have been struggling to round up enough party support for a plan to repeal and replace Obama's health care law. They are now racing against the clock; after Sept. 30, the chamber's rules change and lawmakers won't be able to pass a new bill with just 50 votes. This is critical, since past health care efforts to overhaul health care have failed even with 52 Republicans in the chamber.

"We will have the votes for healthcare but not for the reconciliation deadline of Friday, after which we need 60," Trump tweeted.

Trump also cited support from an unnamed senator that triggered a mystery on Wednesday. "With one Yes vote in hospital & very positive signs from Alaska and two others (McCain is out), we have the HCare Vote, but not for Friday!" Trump said over Twitter. Later, speaking to reporters, Trump again insisted that one of his supporters in the Senate "can't vote because he's in the hospital."

White House officials speaking under condition of anonymity said they believe Trump was referring to Republican Sen. Thad Cochran with his references to a hospitalized senator.

Cochran said he was at home in Mississippi convalescing, but confirmed he was not in the hospital via Twitter: "Thanks for the well-wishes. I'm not hospitalized, but am recuperating at home in Mississippi and look forward to returning to work soon."

The president has also called for scrapping the Senate filibuster rule. It allows a party, in this case the Democrats, to block legislation and force opponents to come up with 60 votes to override the filibuster

As for health care, Trump's prospects for negotiations with the Democrats remain unclear.

Throughout the health care debate, Democrats have said they will not support any plan that makes it harder for people to buy health insurance, effectively cutting them off from coverage. They said all of Trump's plans would do just that.

Sen. Patty Murray, D-Wa., ranking member of the Senate Health, Education, Labor, and Pensions Committee, said Trump can show he is "truly serious" about negotiations by taking "partisan Trumpcare efforts" off the table.

“It would be a welcome shift for President Trump to stop his health care sabotage and instead work across the aisle to help strengthen families’ health care and lower their costs," Murray said.

Contributing: Nicole Gaudiano

More:Trump cites support for health care bill from mystery senator in hospital