Countries including Saudi Arabia, Turkey, Egypt and Israel have said they will become members of the board, and at Davos, President Trump said Putin had accepted an invitation to join the initiative.
But President Putin has not confirmed this and earlier he said his country was still studying the invitation.
Speaking to the BBC’s Breakfast programme from Davos, Cooper said the UK had received an invitation to join the board and strongly supported Trump’s 20-point plan to end the war in Gaza.
“That’s why we are also clear we want to play our part in phase two of the Gaza peace process,” Cooper said.
But she added: “We won’t be one of the signatories today because this is a legal treaty that raises much broader issues.
“And we do also have concerns about President Putin being part of something that’s talking about peace when we’ve still not seen any signs from Putin that there will be commitment to peace in Ukraine.”
She said Putin had shown no willingness “to come and make that agreement and that’s where the pressure needs to be now”.
“But we will have continuing international discussions including with our allies,” the foreign secretary said.
Cooper echoed other UK cabinet ministers who in recent days have been expressing concerns over President Putin’s potential role on the board, given Russia’s ongoing invasion of Ukraine.
As talks to end the war in Ukraine continue, President Trump and Ukrainian President Volodymyr Zelensky are due to meet in Davos on Thursday.
On Wednesday, Trump repeated his oft-stated belief that Putin and Zelensky were close to a deal.