Aberdeen and several English teams are also interested in the 23-year-old free agent, who was on trial at Reading earlier this month without being offered a contract. Tynecastle officials admire his talent and could tempt him north with the prospect of European football this season.
Davenport is a left-footed central midfielder known for his industry, aggression and ability to dictate play from deep. He was released by Blackburn when his contract expired on June 30 and is now seeking a new club.
Hearts hope to hold talks with the player as they try to reinforce the middle of their midfield. Beni Baningime is sidelined until later this year with a cruciate ligament injury and Cammy Devlin is just back after a hamstring issue.
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Full-back Michael Smith played in central midfield alongside Peter Haring on Saturday as the Edinburgh club opened their Premiership campaign with a 2-1 win over Ross County at Tynecastle.
Davenport played in Manchester City's youth academy as a teenager and was loaned out to Burton Albion in 2018. He then joined Blackburn on a four-year contract that summer. Injuries prevented him cementing a regular place in the Rovers side and he made only 36 appearances in total for the Lancashire club.
Hearts would still like up to three more summer signings following the arrivals of winger Alan Forrest, defenders Lewis Neilson, Kye Rowles and Alex Cochrane, midfielder Jorge Grant and striker Lawrence Shankland.