The Tigers are said to have knocked back the Clarets' latest offer on Tuesday evening, shortly before the 26-year-old featured in the 3-2 defeat to Wolverhampton Wanderers.
Slutsky is desperate not to lose any more of his key players before the transfer window closes, admitting that he is "nervous" about the current interest in star man Clucas.
"For me it is very important to keep a player like Sam Clucas because he is a leader," the Daily Mail quotes him as saying. "We spoke with him about his situation but he's a really very professional guy and he has a long-term contract with the club.
"We built the team in a very difficult time. One more out can break the whole team, I'm very nervous about the situation."
Slutsky also confirmed that striker Abel Hernandez will undergo a scan on Wednesday after being stretchered off against Wolves with a "very serious" Achilles injury.