Stanislas featured 32 times for the Clarets last season as the east Lancashire outfit won promotion to the Premier League after finishing runners-up in the Championship.
The 24-year-old Londoner was signed by Howe for an undisclosed fee from West Ham when the Cherries boss was in charge at Turf Moor in August 2011.
Stanislas, capped by England at under-19, 20 and 21 level, progressed through the ranks at Upton Park and played in the Premier League for the Hammers.
He provided the assist for Danny Ings to equalise against Cherries at Turf Moor in November and has reportedly been offered fresh terms by the Clarets.
Cherries have also been linked with former Dean Court loan star Max Gradel, who is currently on international duty at the World Cup with Ivory Coast.
Gradel became a firm favourite with Cherries supporters when he enjoyed a fruitful stay in Dorset on a temporary transfer from Leicester City in 2007-08.
He returned to Leicester to help them win League One in 2009 before going on to star for Leeds as they clinched promotion to the Championship the following season.
Gradel, who came close to signing for West Ham in January, joined French club Saint-Etienne for an undisclosed fee in August 2011.
He has a year left on his contract and is rumoured to be a target for a number of Premier League clubs, including newly-promoted Leicester.
Leyton Orient’s highly-rated winger Moses Odubajo is another player to have been linked with Cherries, with Spurs and West Ham also keen on the 20-year-old.