The 29-year-old was the joint second-highest scorer in the English top-flight last season and Arsenal's bid comes at an uncomfortable time for England, whose manager Roy Hodgson has banned players' agents from the team hotel in Chantilly to avoid his players being distracted.
"If Vardy goes then one or two others might want to leave as well," Taggart, who made over 100 appearances for Leicester, told the BBC. "For the good of the whole team and the club it is important Vardy stays."
Vardy scored 24 league goals last season as Leicester won the top-flight crown for the first time in their history, and was voted as the Footballer of the Year by the Football Writers' Association.
"The whole Leicester ethos over the last couple of seasons has been built around people like Vardy," Taggart added.
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England open their Group B campaign against Russia on Saturday. Wales and Slovakia are the other teams in their group.
(Reporting by Nivedita Shankar in Bengaluru; editing by Sudipto Ganguly)
