The partnership would allow the Premier League side to send players to Rovers to help them develop, and also speed up the receipt of work permits - which currently prevent English clubs from signing a number of talented foreigners.
The East London club assessed various clubs to choose which team they wanted to partner with, and Shamrock Rovers were decided to be the best option.
However, the Sun report indicates that Rovers have not yet held any talks with the Hammers concerning a potential agreement, with the Hoops confirming they have not been approached.
This does not mean the deal is off the table, as West Ham could approach the Irish side later this season, it is just that such rumours have not yet come to fruition.
West Ham have been enjoying one of their best ever seasons this year, and they want to ensure their future continues positively by reaching an agreement with a foreign club.
The deal will make it easier to sign teenage talents from abroad, knowing that work permit restrictions can be overcome with ease.