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Town physio Dave Williams told the club website: "Alan was in the 18 for the Preston game but he wouldn't have been risked.
"He's making good progress this week though and worked hard today without any reaction in the hamstring.
"If all goes well he should be back in full training on Thursday and hopefully okay to play aginst Millwall on Saturday."
Ipswich are without loanee striker Ricardo Fuller whose one match suspension has carried over to the Millwall game.
Lee has scored four goals in his six appearances as a Town player.