Everton have been linked with Asmir Begovic, Jefferson Montero, Andriy Yarmolenko and Bailey Peacock-Farrell in the Sunday newspapers.
Roberto Martinez may bid for a new goalkeeper if there is interest in Joel Robles, according to the Sun on Sunday, and has been looking into the chance of signing Begovic for £5million.
Begovic only joined Chelsea in the summer for £8million from Stoke City, but Thibaut Courtois was quickly reinstalled as No.1 at Stamford Bridge despite Begovic impressing while the Belgian was out injured.
Martinez has also looked into the availability of Michel Vorm at Tottenham and John Ruddy at Norwich City, the newspaper adds, while the Sunday Mirror claims Everton are among several clubs tracking 19-year-old Leeds United goalkeeper Bailey Peacock-Farrell.
The Sun on Sunday suggests Montero, 26, is also a target after falling out of favour under Alan Curtis at Swansea City, with Aiden McGeady and Kevin Mirallas both possibly leaving Goodison Park this month.
Celtic are interested in McGeady, while Mirallas has many admirers in England and abroad, the newspaper says.
The Mail on Sunday, meanwhile, claims the Toffees are preparing a bid of £21million to sign Dynamo Kiev winger Andriy Yarmolenko this month, with Martinez fearful of missing out on a player he has long admired if he waits until after the Euro 2016 finals in the summer.
Lukaku and Stones linked with Everton exits
As well as McGeady, Mirallas and Steven Naismith, who could join Norwich City, both Romelu Lukaku and John Stones could be on their way out of Goodison Park, if reports are to believed.
The Sun on Sunday claims Lukaku has made his mind up to leave in the summer and will snub interest from Manchester United to join either Paris Saint-Germain or Real Madrid, while the Sunday Express says Barcelona are preparing a summer offer for Stones.