The Hammers are due to play their final game at the ground on Saturday 7 May, but the send-off may have to be rearranged.
The Hammers are scheduled to host Swansea City on the penultimate weekend of the Premier League season, but if Slaven Bilic's side progress to the semi-finals of the FA Cup a knock-on effect could see that plan ruined.
With the Irons facing Manchester United in the cup quarter-final in March, the fact that one of the two teams will progress to the semi-finals means that the league encounter between the two, which is scheduled for the same weekend as the last-four ties, will have to be moved.
However, if the Red Devils' advanced to the Europa League semi-finals then that commitment would mean that a midweek rearrangement would be near-impossible.
The Sun claims that the league meeting would therefore likely be moved to the middle of the week after the Hammers host Swansea.
The London club are leaving Upton Park for the first time since 1904 to move into the Olympic Stadium for the start of next season.
Bilic's side are looking to give their old home a memorable final season, with the club embarking upon a lengthy FA Cup run and occupying seventh place in the Premier League table, just one point off a guaranteed Europa League spot.