Everton made a dismal start to the 2024-25 Premier League campaign as they suffered a 3-0 loss at home to Brighton & Hove Albion.
Manager Sean Dyche will hope for an improvement when he takes his side to Tottenham Hotspur this weekend.
The English boss is currently very busy with just over a week remaining in the 2024 summer transfer window.
The Merseyside outfit have already brought in the likes of Tim Iroegbunam, Iliman Ndiaye, Jack Harrison and Jesper Lindstrom, but reinforcements are still needed.
The Toffees already have Jordan Pickford as their number one goalkeeper, with Joao Virginia as his deputy. But Andy Lonergan left to join Wigan Athletic earlier this summer.
Begovic was at Queens Park Rangers last season, where he made 45 Championship appearances and kept 13 clean sheets.
The 37-year-old boasts a lot of experience, having played for clubs like Stoke City, Chelsea, Bournemouth and Everton.
His previous stint at Goodison Park between 2021 and 2023 saw him making 10 appearances for the Toffees, with four clean sheets to his name.
Huge competition for places mean Begovic is unlikely to get many first-team opportunities at Everton, but his vast experience could still be valuable for the club.