Andy Burnham said he is not in favour of the change in Labour policy, that will see the party offer support to changes to the benefit outlined in last week's Conservative budget.
Acting Labour leader Ms Harman said her party had to acknowledge that it lost the election because voters did not trust it "on the economy and on benefits".
She said the party would accept the overall household benefits cap and the chancellor's decision to limit support through tax credits and universal credits to two children.
Ms Harman will now discuss the issue with Labour MPs at a meeting this evening.
Premier's Political Editor Martyn Eden said it was "possible but not very likely" that Ms Harman would be overruled.
He said: "She's not departing from a Labour position but she's saying let's be realistic, the voters don't want us because of this and to tell them you've made a big mistake is a foolish thing to do politically.
"But all the major candidates, all but one for the leadership, have disagreed with her. So it is possible she'll be given a hard time at the meeting."
A spokesman for Andy Burnham, a Catholic, said he would oppose the cuts to child tax credits.
He said: "George Osborne did not set out the details of these cuts before the election and he has no mandate for them. David Cameron even denied they would happen.
"We have seen time and time again with this Tory government that the devil is in the detail, and we're still waiting for George Osborne to set out the details on his cuts to tax credits.
"As we have previously said, Andy opposes cuts to child tax credits. These are paid to people who are doing the right thing and working hard to make ends meet.
"Andy will not offer blanket opposition and, where we agree with a Government policy, we won't oppose for the sake of it. But these tax credit changes are regressive, they are wrong, they hit families in work and Andy opposes them."
But Labour leadership contender Liz Kendall has defended the party's interim leader.
She told the BBC: "People said to us 'We don't trust you on the money, we don't trust you on welfare reform'.
"If we are going to oppose things we have to put something else in its place because if we carry on making the same arguments we have done over the last five years we will get the same result.
"We have to put forward a different credible alternative and Harriet was absolutely right to say that."