The two universities are said to be battling it out to agree a deal with the LA Galaxy footballer to appear as a guest speaker at a future event.
Unions do not usually pay guest speakers to appear at talks, but they are allegedly willing to put money towards any expenses needed by Beckham should he agree.
"David may not seem the most likely candidate but the unions are squabbling with each other to get him," a source told The Mirror.
"His career is unique and they think he would be an inspiration to students. Neither wants to back down so the other can snap him up."
They added: "They don't fork out because they believe the prestige is enough. But they are happy to cover expenses."
Beckham is nearing the end of his contract with LA Galaxy, and has been linked with a move to Paris Saint-Germain, among other teams.
David and Victoria Beckham reportedly spent around £35,000 on Christmas trees this year.
> David Beckham: 'I'm flattered that football teams still want me'