It is Brown's troops who are currently teetering on the edge of the relegation abyss, with eight points separating them from safety.
St Mirren are the side they are chasing down, with Danny Lennon's men having hit a serious slump.
Brown has attempted to crank up the pressure heading into the final weeks of the season by revealing that he would be keen to offer a number of Saints players a top-flight stay if they drop and his Dundee team stay up.
He said in the Daily Record: "If St Mirren don't take any points between now and the end of the season, the same team that won the cup could go down in history as one of the worst teams St Mirren has ever had.
"It would be incredible if Dundee win their next three games, but if that happens and St Mirren don't take any points, St Mirren fans will hunt those players for all time.
"I'm trying to create a squad for next season and I see the lad Craig Samson has not agreed terms there.
"I'll be interested in him and all the players at St Mirren if we stay in the Premier League."