Moyes is seeking to strengthen his West Ham side this month as they prepare for a fight to stay in the top-flight.
The former Everton and Manchester United manager took over from Manuel Pellegrini at the turn of the year.
The Englishman was always a key cog in the Scottish manager's side until the latter took over at Manchester United in 2013.
Despite West Ham's eagerness to bring Barkley to the London Stadium, Chelsea are so far believed to be adamant that they wish to keep the player.
Frank Lampard has only used Barkley in seven of the Premier League matches so far, but that's partly down to an injury that kept him out for over a month.
It must be noted that Lampard has normally relied more on the likes of Matteo Kovacic, Jorginho, N'Golo Kante and Mason Mount.
With Euro 2020 looming in the summer, Barkley will no doubt be keen to play as much football as possible to try and force his way into the England squad.
Moyes is reported to believe this may play into West Ham's favour, as he feels they can offer the 26-year-old more guaranteed game time.
