Allardyce has detected a new sense of maturity in the midfielder after he returned from a loan spell at Birmingham to enjoy an impressive pre-season campaign.
Morrison has a chequered past and found himself in trouble both with the police and the Football Association while he was at Manchester United.
But the 20-year-old tweeted on the eve of the new Barclays Premier League season: "I am a changed man. Don't judge me on my past. Respect me for who I am and how I've developed £WordfromRav"
And that has not gone unnoticed by Allardyce, who was confident when he signed Morrison from Manchester United in January 2012 that he could get the best out of the midfielder.
"He does seem to have matured. He's obviously taken a good look at himself and decided that, if he is going to make it as a professional footballer, he needed to be more disciplined in his day-to-day life," Allardyce told the Evening Standard.
Morrison's big challenge will be to prove he can cut it in the maelstrom of the Premier League.
Allardyce continued: "I haven't had a word with him apart from encouraging him about how well pre-season has gone for him.
"His contribution in pre-season has been exciting for us. His big challenge will be when he gets his chance to play in the first team at competitive level.
"When the whistle goes on Saturday it is 100mph from the start. He will have to learn to cope with that but he is really promising and hopefully we look forward to him continuing to develop into a player who will play a part in the first team.
"We will give him the chance if he continues to do the things he is doing and discipline himself."
Morrison has some stiff competition to break into a West Ham starting line-up with the club not short on attacking talent, with Stewart Downing joining players such as Matt Jarvis, Joe Cole and Ricardo Vaz Te at Upton Park.
But Allardyce believes Morrison could join the likes of Phil Jones and Jack Wiltshire in establishing himself as a Premier League player before he hits the age of 21.
"It is still early days for him and it is a very difficult industry to break into as a young player," Allardyce said.
"Probably the last ones were Jack Wiltshire and Phil Jones, who played for me when I was at Blackburn."