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."