He is set to miss the Six Nations, which starts next month. The World Cup gets underway on 18 September.
"I would not rule him out at this stage," Dallaglio told BBC Radio 5 live.
"He will probably be fit and ready by the start of England's World Cup camp."
Dallaglio, who captained England between 1997 and 1999 and in 2004, added: "He is going to miss a lot of rugby, which is unfortunate".
"But let's hope he is back fit and ready to be available come June or July."
Morgan, who has 27 caps, was one of England's most impressive performers during the autumn internationals, scoring a try after coming on as a replacement against South Africa, before being promoted to the starting XV against Samoa and scoring two tries in the win over Australia the following weekend.
He signed a new contract with his hometown club last week, having joined them in 2012 after a spell with Welsh regional side Scarlets.
Morgan made his England debut against Scotland in February 2012, turning down the chance to represent Wales, for whom he would have qualified on residency grounds from his time at Parc y Scarlets.
World Cup hosts England open this year's tournament against Fiji before other Pool A matches against Australia, Wales and Uruguay.