After Spurs' title hopes were severely dented by a 1-0 defeat to West Ham on 5 May, Allen said his former club needed more experienced heads in the dressing room to help navigate high-pressure situations.
Carrick, 35, joined United from Tottenham for £18.6m ($23.2m) in 2006. He is out of contract at Old Trafford this summer, with United yet to offer the veteran midfielder a new deal.
Allen also suggested Spurs pursue Sunderland striker Jermaine Defoe, who has scored 14 goals this season, to add more depth to their forward positions.
"They've stuck with this young group and they have done very well, but I just felt at times they needed some guidance, they needed a leader to get them over the line," he told talkSPORT.
"Ironically for me it would be someone like Michael Carrick... he's been through everything – has had some tough times at Manchester United but won things there. He's a real solid professional who I think could be a massive influence on the group.
"There also needs to be more strength in depth and Jermain Defoe would be another fantastic addition," Allen added.
"At this stage in his career he knows he won't play every game, but he's in great shape and he has a fantastic attitude and I certainly wouldn't deny that he would be a great addition to the squad."
Spurs are four points behind league leaders Chelsea, who have a game in hand. The north London club host United in their last ever game at White Hart Lane on 14 May.
Allen insisted that Tottenham's season would not be a failure if they failed to win the league title.
"There is very little more they could have done to really end up as winners, they've gone so close, but the season will be deemed a success – they've earned qualification to the Champions League," the former striker said.