The 21-year-old attacking midfielder has been on a season-long loan but Hammers chiefs may look at cutting the deal short so he can play in a higher division.
There is hope he can compete for a first-team place next season following a good run in the Championship.
The likes of West Brom, Sheffield United, Stoke, Derby and Bristol City are all understood to be interested in the youngster.
Browne has starred for Karl Robinson's men in League One, where he has banged in five goals in the third-tier.
While he has also chipped in with four assists in 17 games, as the U's sit in 18th spot in the table.
Former West Ham defender Rio Ferdinand has regularly offered Browne advice, with the player speaking of his gratitude to the Champions League winner.
He told the Oxford Mail: "Rio knows my agent really well, so we work closely. We’re in constant conversation about how I’m doing and he gives me advice.
"We do message each other and chat to make sure I’m keeping my head down and staying focused.
“He’s a person with a lot of experience, so when people like that try to help then you can’t do anything but listen and take it on board. It does help."
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