The 27-year-old has played just 22 Premier League minutes this season.
"He has been struggling to get a bit of match-time," said striker Vardy.
Drinkwater, who won the last of three England caps in May 2016, only made his Chelsea debut in late October having struggled with thigh and calf injuries since joining the champions in August.
"He was injured and he needs to get the match sharpness back. He has been doing that in recent weeks for Chelsea and he needs to keep building that up," added Vardy.
"He definitely wants to play for his country."
Southgate said he had spoken to Drinkwater before this international break and said the midfielder "felt he wasn't ready yet to play this level".
The former Middlesbrough manager added: "If Danny didn't feel he was ready for that I can understand it given the amount of minutes he's had and he didn't play at the end of the season with Leicester either."
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