Asamoah, 29, was offered a new contract but snubbed it as he wants a new challenge.
The Ghana international has attracted interest from Inter Milan and Tottenham with Chelsea identifying him as a solid alternative if they fail to sign Alex Sandro.
But according to The Sun, Everton have leapfrogged them in the race to sign the versatile midfielder by making a 'firm approach' for him in the past few days.
Despite this Tottenham are still confident of landing Asamoah and are also reported to have tried to contact his representatives.
The 29-year-old moved to Juventus in 2012 and has won the Serie A title with them for a record six years in a row.
He has represented the Old Lady 150 times, scoring five goals, but has been blighted by injuries the past few years.
Despite having the option to remain in Italy the seasoned campaigner seems increasingly likely to be playing in the Premier League next term.