The pair worked together at Craven Cottage, where Reed became an important part of Parker's midfield.
His style of play is understood to be a key component of the football the new West Ham manager wants to implement, making the 31-year-old an interesting option as he looks to reshape the squad.
Parker's detailed tactical plans were an asset that impressed Reed during their time together, and the midfielder's understanding of his methods could prove valuable if West Ham decide to pursue a deal.
Reed has been at Fulham since 2020 but has yet to feature in the Premier League this season. He has captained the side in the Carabao Cup, although their elimination from the competition means that particular route to regular minutes is now closed.
A move could therefore offer the midfielder a fresh opportunity, while giving Parker a familiar presence who understands exactly what he expects from his players.
Parker, 46, has signed a two-and-a-half-year contract at the London Stadium after previously managing Fulham, Bournemouth and Burnley. He arrives with West Ham needing to get their season back on track after a disappointing start.
The Hammers were among the favourites to win the Championship but currently sit eighth in the table, leaving them with work to do if they are to meet expectations.
Parker's immediate priority will be to restore momentum, but the January window will also give him an opportunity to put his own stamp on the squad. Reed's familiarity with his approach could make him one of the names to watch as West Ham look to strengthen.