Deeney has been continuously linked with a Premier League switch this summer following another impressive campaign for Watford in the Championship, and Burnley have now stepped up their interest in the forward.
The 26-year-old scored 24 goals in 44 games last season and the two promoted clubs are now looking to bolster their attacking options by adding Deeney.
Leicester have reportedly had a £7.5 million bid rejected and have yet to go back with another offer, though Burnley are believed to have had an £8 million bid turned down as well.
The Hornets are holding out for a fee in excess of £10 million for their newly-designated captain, with QPR and Aston Villa also interested.
Leicester have already signed Leonardo Ulloa from Brighton for around £10 million and may be reluctant to spend big on another forward from the Championship.
Burnley will hope to benefit if this is the case and if they can improve their offer, Sean Dyche could be reunited with Deeney, having worked together while Dyche was Watford manager.
The Watford striker is ready for life in the Premier League after scoring 45 goals in his last two campaigns and will relish the challenge if a top-flight side can match Burnley's valuation.