Laws offered the 34-year-old a one-year deal with the option of a further 12 months based on appearances, but Blake was unhappy with the terms.
But Laws will leaving the door open for the veteran, who has made over 200 appearances for the Clarets spread across two spells.
"Obviously I am disappointed that Robbie has turned down what we consider to be a very reasonable deal for a 34-year-old," Laws told the club's official website. "The offer is a 12-month deal with a further one year option but Robbie wanted two years and we just couldn't go down that road.
"We included the extended deal should Robbie play 20 games next season and I feel a fit Robbie would have easily achieved that. So naturally I am disappointed, but would like to thank Robbie for all his magnificent service for the club.
"I know he is a firm fans' favourite and he is rightly considered a legend here, who will never be forgotten. But I have told Robbie and his agent that we have left the door open and if he has a change of mind I would be more than happy to talk."