The 20-year-old recently signed a new contract with City but has only made four appearances for the first team this season, scoring in the 3-0 win over Arsenal in the Carling Cup quarter-final.
Wigan and German side Cologne were also linked with a move for the player, who is hoping to gain more first-team experience in a bid to ensure he is part of Slovakia's World Cup squad this summer.
"Vladimir is an exciting young player who adds further options to our squad and can help to get the fans out of their seats," Coyle told the club's website.
"The fact that he has a chance of going to this year's World Cup shows just how much ability he possesses."
Weiss came up through the youth ranks at City and made his senior debut as a second-half substitute in their final game of last season - ironically, against Bolton.
The highly-rated winger has been a regular on the bench for City this season, but has only been given playing time in the cup competitions.
Bolton 4/5, Draw 5/2, Burnley 7/2