Blake rejected the offer, which included the option of another year based on appearances, as he wanted to secure a straight two-year deal.
Laws remains keen to keep Blake at Turf Moor, though, and says the original terms are still available to him.
He told the club's official website: "I know Robbie Blake is a huge favourite with supporters. It has already been clearly stressed that the offer is still on the table and I hope that Robbie is still thinking about it.
"We want Robbie at Burnley Football Club. He is a crowd favourite. I don't think he will get that anywhere else and we know what he can give us. At the moment it's a bit of a stalemate but there's still plenty of time and we will give him that time."