The 26-year-old injured his arm in the 4-1 defeat at St Mary's Stadium and Lambert confirmed that the problem is likely to keep him out for a number of weeks.
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"It's unfortunate for Stephen and it's unfortunate for us," Lambert told the Express and Star.
"I don't think it's as bad as we thought [though]. Hopefully he'll be back in a few weeks."
Villa had recovered well from their comprehensive defeat on the south coast, beating Manchester City 4-2 after extra time at the Etihad Stadium to secure their passage into the fourth round of the League Cup.
But Lambert's side face a tough test against neighbours West Brom, who have made a good start to the season so far and sit fourth in the table.