So if their positions are so similar, why has it been so difficult for the city's most powerful men to get together?
An optimist would say it's simply a strategic disagreement that will resolve itself: Nutter and Clarke have different philosophies on how to get things done in GOP-controlled Harrisburg.
A pessimist would say that it's about personality, that one or both of them are picking a fight to make the other look bad.
"I'm hoping that they don't get into an ego dispute, like, 'I didn't come up with the solution, so I can't come along with it,' " said Phil Goldsmith, a former city managing director under Mayor John Street and former school- board member. He was the school district's interim CEO in 2001.
"Unless we've gotten rid of egos in Philadelphia, that seems to be possibly what's going on."
So what exactly are they fighting about?
The state in June authorized the city to make permanent a temporary sales-tax increase (from 7 percent to 8 percent) as part of a plan to plug the district's $304 million deficit. That plan would send the first $120 million in revenue to Philly schools, with whatever's left, after possible debt payments, going to the pension fund.
In an ideal world, both Nutter and Clarke say they would prefer a balanced split between school and pension aid. Here's where the difference in strategy comes in.
Nutter wants the city to bite the bullet and take what's been given to it by Gov. Corbett and state lawmakers. Doing so, he says, will preserve the city's good standing in Harrisburg so it can work to modify the sales-tax language and pursue other goals, like the funding formula.
Clarke has said Council will reject the sales tax as written and wants the city to take to the bully pulpit to demand a package that better addresses Philly's needs.
He also opposes a provision that requires the state secretary of education to certify that financial and operational reforms are underway in the district before releasing the $120 million.
"City Council has been clear: The funding package offered by the Corbett administration in June is unfair to the school district and to Philadelphia taxpayers," Clarke spokeswoman Jane Roh said in a statement. "There is no good reason to threaten both the school district and Philadelphia taxpayers with unreliable funding year after year."
Tim Eller, spokesman for the state Education Department, said the governor will wait to see what Council does before considering any changes.
Supporters of Nutter and Clarke accuse the other side of naivete - and both sides could be right. Many sources familiar with Harrisburg say they are skeptical the state will modify the sales-tax authorization, whether Council approves it or not.
"What sort of string of victories or successes in Harrisburg has Clarke had?" a top administration official asked. "I don't think we can have a lot of confidence in his ability to make it happen."
Former Democratic state Rep. Dick Hayden, who played a critical role in Nutter's 2007 mayoral campaign, said Clarke's strategy won't fly when Republicans control the House, Senate and governor's mansion.
"The mayor's approach recognizes the realities on the ground in Harrisburg," said Hayden, a lawyer and lobbyist who works in Philly and Harrisburg. "It reflects a certain naivete to think that just the plight of the city of Philadelphia or the school district has a lot of resonance in Harrisburg."
State Sen. Anthony Williams said Clarke's strategy is shaky because Corbett has so little control over the General Assembly that even changing his mind through public pressure won't guarantee action.
"Darrell's strategy is to hold out for a better deal. It may work with an administration that has the ability to do something," said Williams, a Southwest Philly Democrat. "It's the House of Representatives which has not cooperated with the governor, and certainly its leadership doesn't care a great deal about the city's schools."
At the same time, if Harrisburg is truly unwilling to change the tax, Nutter would be stymied in his plan to lobby after the fact for splitting the revenue 50/50 and removing the requirement that the district's reform efforts receive annual state certification.
City Controller Alan Butkovitz, a former state representative, compared the situation to a union president who reaches a deal with management, only to have it rejected by membership.
"Nutter was the ambassador or the representative of Philadelphia to the negotiations," Butkovitz said. "Darrell Clarke is saying that you have to go back to the negotiating table."
(Nutter has said that although he, like Clarke, went to Harrisburg several times last budget season, the governor orchestrated the final deal without him.)
Butkovitz said he agrees with Clarke's position that the city shouldn't go along with a bad plan that comes from Harrisburg.
"You have to fight, you have to negotiate, you have to make allies and you have to make it difficult for Harrisburg," he said.
On Twitter: @SeanWalshDN