The current Gunners boss is coming under a lot of pressure following a difficult run of results in all competitions.
Arsenal have squandered a lead in their last four games in the Premier League, the Carabao Cup and the Europa League.
The Gunners have only managed to win one of their last six games following a 3-2 victory over Portuguese outfit Vitoria de Guimaraes.
Arsenal drew 2-2 with Crystal Palace before a 1-1 stalemate with Wolves in their last two Premier League fixtures.
Emery’s side are in fifth spot in the Premier League table and six points adrift of third-placed Leicester City.
The Foxes, in contrast to Arsenal, are on a rich run of form following a return of 23 points from 11 games.
Leicester were 9-0 winners against Southampton at St Mary’s last month before the Foxes beat Crystal Palace at Selhurst Park last weekend.
Former Spurs manager Sherwood believes Rodgers is the clear choice to replace Emery if Arsenal decide to go for a change of management.
“They might not have to go far to negotiate with another manager because he might be sitting right next to him [at the weekend],” Sherwood told Sky Sports.
“I think Brendan Rodgers is an obvious choice for a top, top Premier League team.
“Would he want to leave Leicester? I’m not saying he would. But if I was at the head of it at Arsenal, I like Brendan Rodgers, I think he’s a better manager than Unai Emery. I would certainly be trying to tap him up.”
Arsenal haven’t won the Premier League title since 2003-04.
The Gunners have missed out on a place in the top four in the last three Premier League seasons.
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news