Steven Lawless has been given a new deal by Partick, a few days after being handed a ban for breaching betting rules.
Midfielder Miguel Pallardo has signed a new one-year deal with newly-promoted Hearts.
This has led to intense speculation that Moyes could be set to bring his Goodison Park reign to an end after more than a decade in charge.
"We haven't done too bad in the ten years or so that he has been here. It's not like we have under-achieved. If anything, we have slightly over-achieved in terms of budgets and transfer funds," said Jagielka.
"We don't have to feel that we have to prove anything. The manager has pretty much brought every single player in, bar one or two, so we are his players.
"He has made decisions that he wants to make. He has come out in the press and said he will not sign anything until the end of the season, which is fine with us."
Jagielka was talking after Everton's 2-2 draw with Oldham, which saw them into the FA quarter-final draw where they could face Wigan at home.
"Obviously, it will be nice to give him a quick reminder of how good we are by keeping this cup run going and pushing as high as we can up the Premier League," he said.
"At the end of the day, if we do win the cup and he doesn't sign, it will still be a fantastic feat for the players.
"We are a very close squad and we get on very well. As much as we go out there to do it for the manager, we do it for each other as well.
"Obviously, the fans, too. We had a good following today and the fans are brilliant, too. So there is more than one person there to play for."