Bosnich, a two-time League Cup winner with Villa, reckons Unai Emery has the club on course to become serious Premier League title contenders in the coming years and hopes Martinez still has a huge part to play.
The Argentina international fuelled speculation over his future by making a tearful exit from the pitch following Friday’s 2-0 home win over Tottenham.
Clubs from Saudi Arabia and Europe are thought to have approached the keeper’s representatives in recent weeks.
Bosnich told Sky Sports: “I know they have the PSR rules (to consider) but I would caution, not Emi, but the club to be very, very careful because replacing someone in such an important position is easier said than done.
“I think really both himself and the club have unfinished business. The way they are going and projected to be going, they are going all the way.
“By that I mean they are going to be serious title challengers within the next couple of years.
“I would think very, very long and hard. Unless you have a ready-made replacement, which I would find unbelievable because we are talking about arguably the best goalkeeper in the world right now.
“Or unless behind the scenes we don’t know what is happening and Emi has said he is ready for a new challenge. If that is the case it is a different story altogether.
“But please, I would say think very, very carefully before you start tinkering with someone who has been such an important part of Aston Villa’s recent journey.”
Martinez has made 211 appearances and twice won the Lev Yashin Trophy, awarded to the best goalkeeper in the world, since joining Villa from Arsenal for £20m five years ago.
Asked for his reaction after seeing the 32-year-old’s emotions on Friday, Bosnich replied: “It really stirred something inside of me because I think he has been absolutely fantastic for Aston Villa.
“He has been a major part of Villa’s resurgence, not only in the league but also in Europe as well.
“For me he was the final bit of the jigsaw for Argentina to go and win the World Cup.
“The first thing I thought was: ‘Is he OK?. Then I started thinking about it and digesting about it and it did seem like a goodbye.”