Dragovic only played 16 times for Leicester last season, but made a positive impression on the Foxes fanbase.
The newspaper claims Claude Puel's side have an option to sign the Bayer Leverkusen defender outright, but adds that their failure to conclude a deal has alerted other clubs.
And Everton, West Ham and Brighton and Hove Albion are all said to be monitoring the situation.
According to the Leicester Mercury, Dragovic is expected to cost around £15 million.
The Daily Mirror reports West Ham are edging closer to agreeing a £20 million deal for Swansea City's Alfie Mawson, which could open the door for Everton or Brighton to sign Dragovic.
Leicester have already strengthened their options in the Austrian's position, landing Jonny Evans from West Bromwich Albion for around five times less than the rumoured Dragovic asking price.
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