Ravel Morrison has returned to parent club West Ham from Cardiff City on compassionate leave.
Brighton boss Sami Hyypia is weighing up a move for former France captain Alou Diarra.
Stock's move to Turf Moor from Doncaster on a two-year deal reunites the 30-year-old with former Bournemouth team-mate Howe.
Stock is Howe's sixth summer signing and the Clarets chief told the club's official website: "Brian is a very good footballer who will suit how we play.
"He is someone we admired for a long time. He had a really good career at Doncaster and I thought he was impressive in the games he played against us over the past couple of years.
"Experience and leadership are two things we haven't got a great deal of and I certainly think he will add in those two areas."