Transfer Gossip: Pompey interested in League One midfielder
Barry Fry told the Peterborough Telegraph that the Blues are one of several clubs tracking the 31-year-old.
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Hide AdBlackpool, Shrewsbury and Lincoln also reportedly have Grant on their radar.
Fry revealed the Posh have given the London-born talent permission to speak to a club, although a fee has yet to be agreed.
The director of football said: 'Numerous clubs have shown an interest in him including Lincoln, Shrewsbury, Portsmouth and Blackpool.
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Hide Ad'Anthony has been given permission to speak to a club without a fee yet being agreed.'
Grant made 48 appearances for the Posh last season, helping them to a ninth-place finish in League One.
He graduated through Chelsea's academy in 2004 and made a solitary appearance for the Premier League outfit in a 3-1 win over Manchester United in April 2005.
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Hide AdGrant has also represented Southend United, Stevenage, Crewe and Port Vale during his career.
He has a year left on his Peterborough contract after moving to London Road for a six-figure fee in January 2017.