The chairman of the House HELP Committee signaled that the House could pass the Senate bill as-is before considering a reconciliation measure to “correct” the bill, a key concession to Senate Democrats who wanted the House to act first:

Rep. George Miller (D-Calif.), the chairman of the House Education and Labor Committee and a close ally of Speaker Nancy Pelosi (D-Calif.), suggested that the House may pass the Senate bill before any other action is taken — a key concession to Senate Democrats.
“The choreography gets a little complicated here, but the House will present a reconciliation bill,” Miller said during an appearance on MSNBC. “It will be based on many of the principles that the president put forth to correct some of the problems the House and others have had with the Senate bill.”
[. . .]
“That may require us to pass the Senate bill first and then send the reconciliation bill to the Senate for them to pass,” Miller said. “I think Sen. Reid believes he can put the votes together for that.”
[. . .]
“That means that I would hope we get it done this month,” he said. “And hopefully the leadership will come together and figure out where we are as a result of the summit yesterday.”