The Toffees are keen to sort out the futures of a number of players after securing their Premiership safety.
Unsworth, who is out of contract at the end of the season, rejected the club's first offer of a new one-year deal as he is holding out for a two-year extension.
Bolton, Rangers and Manchester City are all thought to be monitoring the defender's situation, but Everton are keen to keep him and have offered him a two-year deal.
Radzinski's future at Everton has also been in doubt with Blackburn and Portsmouth linked with the Canadian international.
The pacey striker has 12 months to run on his current deal and the club have offered him a two-year extension.
Toffees boss David Moyes is now waiting to hear from both players on whether or not they accept the offers.
"We've offered Tomasz Radzinski a new deal as we have done with David Unsworth," Moyes told the club's official website.
"We've offered them both two-year contracts. It's just a case now of waiting and seeing. We've made those offers and we'll wait and see what happens."
As well as trying to keep players on new deals Moyes is working on bringing in new faces for next season.
"We're always ready and looking for things that can make the squad better. Everybody knows that finances are tight and that decision ultimately can't be made by me," added Moyes.
