Arsenal are said to be keen on pursuing Ipswich Town's Tyrone Mings instead, meaning Liverpool are favourites to move for the Hammers man.
Reid's contract expires at the end of the season but the People report that he wants a move away from Upton Park - with Liverpool in pole position as things stand.
Meanwhile, West Ham boss Sam Allardyce says he is 'comfortable' with how contract talks are progressing with the club:
"I think it will happen like it did last time when we get closer to the end of the season," Allardyce told reporters. "We will decide on where we are going and what we are going to do.
"Just after getting promoted [in 2012] there were no new contract talks in the Premier League until April and the back of end May. It will probably be the same this time round. I’ve always been comfortable with that. I was last time and I’m comfortable with it this time."