Vine, who accidentally copied a member of the public into the email, says that media mogul Rupert Murdoch and Daily Mail editor Paul Dacre also have reservations about Johnson.
Vine urges her husband to seek "specific assurances" what his role might be in a cabinet led by Johnson.
The email was passed to Sky News.
It reads: "Very important that we focus on the individual obstacles and thoroughly overcome them before moving to the next. I really think Michael needs to have a Henry or a Beth (Mr Gove's media advisers, also copied into the email) with him for this morning's crucial meetings.
"One simple message: You MUST have SPECIFIC assurances from Boris OTHERWISE you cannot guarantee your support. The details can be worked out later on, but without that you have no leverage.
"Crucially, the membership will not have the necessary reassurance to back Boris, neither will Dacre/ Murdoch, who instinctively dislike Boris but trust your ability enough to support a Boris Gove ticket.
"Do not concede any ground. Be your stubborn best.
Johnson is one of the favourites to take over from David Cameron as Tory leader.
Gove and the former London Mayor joined forces to campaign to Leave the European Union.