The Midlands club faced an uncertain future after having to sell the majority of their first-team players.
It finished off his father Brian and his own connection with the Midlands club.
The offer is believed to be worth well over £1 million, though the Midlands club will face competition from big Premier League clubs.
Burke, who joined the Midlands club two years ago for £15,000, has had an inconsistent season.
He played a total of 39 games, and scored 4 goals for the Midlands club.
Ireland signed a 4-year deal with the Midlands club.
Leicester were playing in a regional league organised for Midlands clubs in this period.
In total, he made 194 appearances for the Midlands club, scoring 16 times.
He is signed to the Midlands club until Summer 2013, with the option of a further year.
He signed for the Midlands club, leaping at the chance to play in the First Division on a regular basis.