Premier League new boys Fulham are seeking to sign their two on-loan midfielders for next season.
The pair, who spent last season on loan from Southampton and Bournemouth respectively, are both firm favourites of Parker and keen on a stay at Craven Cottage.
Parker himself is set to extend his deal with the club having secured promotion, with talks to begin when they return from short breaks. The 39-year-old and his coaching staff were handed two-year deals in May 2019.
Reed has informed Southampton of his desire for a regular starting place, and while manager Ralph Hasenhuttl is an admirer of the 25-year-old, he cannot give such assurances despite the expected loss of Pierre-Emile Hojbjerg to Tottenham.
Fulham remain in talks with the south coast club but are confident of sealing a deal worth around £8million for the tenacious midfielder.
Arter's loan deal included an obligation to buy for £4m had he made a certain number of starts. Despite that number not being met, Fulham remain keen on striking a deal.