For Rice I seem pretty sure about Arsenal but for Romeo Lavia I still haven't heard too much of these rumours because as far as I know he is one of the players that Chelsea and Liverpool are fighting over but not Arsenal.
But it looks like West Ham have turned down Arsenal's offer to take Rice with them next season.
Rice is a player who can provide a lot of good development for West Ham so it is only natural that an offer worth £ 80 million was rejected by West Ham.
Not only you, I also haven't heard too many rumors that Arsenal are eyeing Romeo Lavia, a player with a midfield position from the Southampton team.