Conservative allies publicly supported Boehner's plan Wednesday.
Anti-tax crusader Grover Norquist provided political cover for Republicans who have signed his pledge against tax increases, saying he could support plan B.
Obama and Democrats argue that increased revenue, including higher tax rates on the wealthy, must be part of broader deficit reduction plan.
Obama made the tax proposal a theme of his re-election campaign, arguing that it would prevent a tax increase for middle-class Americans.
Polls show support for the Obama plan, and some Republicans have called for acceding to the president on the tax issue in order to focus on cuts to spending and entitlement programs.
Boehner and Republicans initially opposed any rise in tax rates but agreed to raising revenue by eliminating some deductions and loopholes. The offer of a plan with higher rates for millionaires represented a further concession, but Obama and Democrats say it would not suffice.
Senate Majority Leader Harry Reid of Nevada said Boehner's plan appeared to be a result of pressure from tea party conservatives opposing a wider deal.
"It would be a shame if Republicans abandoned productive negotiations due to pressure from the tea party, as they have time and again," Reid said this week.
Boehner's spokesman, Michael Steel, shot back that the plan B proposal gave Democrats what they wanted -- higher tax rates on millionaires.
Obama's latest offer has generated protests from the liberal base of the Democratic Party because it includes cuts in entitlement programs such as Social Security, Medicare and Medicaid.
Justin Ruben, executive director of MoveOn, which backed Obama's presidential campaigns, said its members would consider any benefit cuts "a betrayal that sells out working and middle-class families."
In particular, liberals cited concessions that Obama made Monday in his counteroffer, including a new inflation formula applied to benefits called chained CPI.
Chained CPI includes assumptions on consumer habits in response to rising prices, such as seeking cheaper alternatives, and would result in smaller benefit increases in future years.
Statistics supplied by opponents say the change would mean Social Security recipients would get $6,000 less in benefits over the first 15 years of chained CPI.
But White House spokesman Jay Carney said Obama's CPI proposal "would protect vulnerable communities, including the very elderly, when it comes to Social Security recipients." He called the president's acceptance of the chained CPI a signal of his willingness to compromise.