The 32-year-old picked up a thigh injury against Mansfield last week but his manager is confident he will recover in time to be available for his side's Premier League opener
Former England midfielder Cole, who arrived at Villa Park as a free agent last month, sat out Wednesday's 2-0 win over MLS side Dallas after being hurt in a friendly against Mansfield last week.
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However, Lambert is confident the 32-year-old will be ready to make his competitive debut when Villa begin the new season at Stoke City on August 16.
"Joe has a little knock on his thigh. It's a minor knock," Lambert told Villa's official website.
"He will be touch and go for Saturday. I won't rush him.
"But I am not too concerned. I think Joe will be fine for the first day."
A major positive for Villa has been provided by the return to fitness of winger Charles N'Zogbia, who missed the entire 2013-14 campaign due to a serious Achilles problem.
N'Zogbia scored a spectacular free-kick against Dallas and Lambert said: "It is great to see Charles come back after a lengthy spell out - it was a bad injury.
"He's come back in pre-season looking pretty sharp. If he can stay injury-free, hopefully he will have a big season.
"Overall his contribution since he's been back has been very, very good. His goal was excellent. To come back the way he has is very pleasing."