The Merseyside club are looking for a new manager for the first time in 11 years following David Moyes' departure to Manchester United as Sir Alex Ferguson's replacement.
Cardiff boss Malky Mackay and Porto's Vitor Pereira have also been linked with the vacant post, as have former Everton players such as David Weir, Phil Neville, Duncan Ferguson and Alan Stubbs.
But Wigan manager Martinez is the bookmakers' favourite to take the reins at Goodison Park, with talks already under way.
Everton chairman Bill Kenwright has stressed the Spaniard is not the only option, but his appointment would be welcomed by some of the club's key players.
The Everton captain, currently on international duty with England, said: "I will be delighted.
"Everton is a massive club and hopefully we can get a fantastic manager."
His Everton team-mate Baines, also away with England, said: "It all sort of seems to be going on at the moment.
"Thankfully, we are in camp so I am keeping it at the back of my mind for now and concentrating more on England.
"Roberto has been linked with the job. He is a terrific manager so we will just wait and see."
A decision on Moyes' successor could have well been made by the time the pair return from South America, where England face Brazil in a high-profile friendly at the Maracana on Sunday.
If England are to get a result against the 2014 World Cup hosts, though, they will need to play better than they did against Ireland last night.
Hodgson's side were made to look decidedly ordinary by a side 32 places lower down the Fifa rankings, with Frank Lampard's first-half strike earning a draw.
But Jagielka was keen to draw the positives from the game, insisting it will help in their World Cup qualifiers against the likes of Ukraine, Montenegro and Poland.
"I am sure the fans enjoyed the game," the defender said. "It was pretty much end to end.
"Unfortunately for us we did not manage to get the win, but there are still quite a few positives to take out of it.
"If you look at the games we have left in the group, especially at Wembley, teams will try to frustrate us.
"After getting the goal, that was Ireland's gameplan – to try and hit us on the counter with longer balls forward.
"They did it well and managed to keep it down to one. We had quite a few opportunities to score a goal or have a killer, final ball, but we will still take positives from the game."