Wednesday would have marked the first time DUP MPs voted to pressure ministers since their party agreed to a confidence and supply arrangement with the Conservatives after the general election.
The Tories are now expected to sit in their hands in the two votes - allowing the motions to pass in the belief they do not bind the Government to act.
May herself was absent and decided not to put the issue to a vote after the Democratic Unionist Party (DUP) had signalled it would back the Opposition Day motion.
He said: "The consequences of losing financial discipline for a government are not just pay freezes and one per cent caps, but a million unemployed as a result of that recession we had post 2008".
"We will support them if this matter goes to a vote tonight, and it's interesting to see if we will actually get to that point".
Mr Paisley said: 'To those members of the Labour Party who chide about the £1 billion deal, your party would quite happily have cut a deal that would probably have been better for us.
Political Correspondent John Manley
Today is the first time since agreeing a confidence and supply agreement to prop up Mrs May's minority government the DUP have moved against the Tories.
"We made it very clear to her Majesty's government on issues like this we reserve the right to vote on the basis of our own manifesto".
Conservative sources insisted they were "pretty relaxed" about the outcome of the debate, which does not require the Government to change policy.
Jonathan Ashworth, Labour's shadow health secretary, said the Government's move was exclusively aimed at avoiding defeat - and urged Mrs May to scrap the pay cap for all public sector workers.
Health Secretary Jeremy Hunt said that the motion calling for an "end [to] the public sector pay cap in the NHS and [to] give NHS workers a fair pay rise" was "bogus".
It comes the day after ministers effectively ended the pay cap with the announcement of rises above the 1% limit for police and prison officers.
"It's extremely rare for the Government not to vote down an Opposition motion and the only explanation is it avoided a vote because it knew it would lose it", he said.
DRBC's partial ban on fracking a step backward
New York, Pennsylvania and DE voted yes on the ban, New Jersey abstained from voting and the federal government voted no. We will keep fighting to protect the basin from fracking and fracking waste.