Senate Democrats on Monday pushed Senate Majority Leader Mitch McConnell to work with them on an Obamacare fix as Republicans returned to Washington facing growing doubts that they can pass their bill to overhaul the health-care system.

In a letter to McConnell, Minority Leader Chuck Schumer and three other Democrats called on McConnell to "focus on immediately advancing policies to provide stability and certainty to the health insurance markets." They asked Schumer to work with them in "advancing measures that would have an immediate impact on improving the health care system for American families."

Returning from a Fourth of July break with only three weeks until a recess for August, Republicans sit multiple votes short of the support needed to pass the Better Care Reconciliation Act. The Senate GOP is working on changes that could win over both its conservative and moderate wings, but those factions hold sometimes competing concerns about what it takes to improve the plan.

Sen. Pat Toomey, R-Pa., told CNBC on Monday he expects a revised version of the bill as early as Monday and still sees "a shot" of reaching the 50 votes needed to pass a plan. A spokesman for McConnell told CNBC that he does not "have any expectations of a bill release today," though he noted that the majority leader has said a Congressional Budget Office score is "forthcoming."