Among those reported to have a place on the board are Lubov Chernukhin, a former banker and the wife of Putin’s former deputy finance minister Vladimir Chernukhin.
It comes as the UK and other Western nations threaten Putin with severe economic sanctions if Russia invades Ukraine.
Asked about the Sunday Times report, shadow health secretary Wes Streeting said the government “has got serious questions to answer”.
“Donors have an enormous degree of influence over the policies and priorities of the Conservative government,” he told Sky News’ Trevor Phillips on Sunday programme.
“There are questions… to answer on this and it’s important the public is sure the government is making the right decisions for the right reasons.”
He added: “We’re particularly concerned in the context of our discussion this morning about those individuals with links to president Putin’s regime, but this applies more broadly in terms of our politics, which is that it can’t just be big money donors that call the shots in our democracy.”
Streeting also accused the government of failing to take steps to “clean up our act at home” regarding Russian money in London in particular.
He said the government had not paid sufficient attention to a cross-party report by parliament’s intelligence security committee which accused the government of not doing enough to investigate or protect the UK from possible Russian interference in the 2016 Brexit referendum.