Blackman struck a hat-trick in the 4-0 Clydesdale Bank Premier League win over St Johnstone at Fir Park.
The forward took his tally to eight SPL goals from 10 starts on the eve of his 21st birthday and Brown is eager to keep him after his loan deal expires in January.
Brown was delighted to hear Blackburn had also won, at Newcastle, and hopes to capitalise on his colleague's good mood.
"I always phone him (Allardyce) after he wins and tell him he doesn't need another striker," the former Scotland manager said.
"We would be to anxious keep him. I look for the Blackburn result every game because, the more they win, the more chance we have of keeping him.
"He wants to stay, we're hopeful. Sam's been very good with me.
"The deal was arranged in a swimming pool in Qatar in the summer, we were both commentating on the World Cup for Al-Jazeera."