Sunderland had previously shown interest in Petrovic, and Monaco were also linked, but the Cherries are now frontrunners to sign him.
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Petrovic is keen to join Bournemouth, and there is confidence on all sides that the move will be completed.
Chelsea are currently planning to go into the new season with Robert Sanchez and Filip Jorgensen as their two main goalkeepers, and have been open to moving Petrovic on following a season-long loan at sister club Strasbourg.
Petrovic was named Strasbourg’s player of the season and is keen to remain a first-choice keeper.
The Serbia international signed a seven-year contract with Chelsea when he joined from Major League Soccer side New England Revolution in 2023.
Bournemouth have been exploring options for a new No1 after losing Kepa Arrizabalaga. The Spaniard was their first-choice goalkeeper last season but, after returning to Chelsea at the end of that loan deal, has joined Arsenal after his £5m release clause was triggered.