So Lazio could have asked his agent if he's willing to move, your GK said yes and Lazio knew he would throw a fit if you rejected their bid for him. If your GK didnt care about the rejected bid it would've been higher - to sway you to sell rather than sway the GK to move.
If I was Lazio, I would do the same thing.
BTW, the game takes quite some time to recognize your success, and see your team as being "permanently" improved rather than just an overachieveing Burnley that could go on a slump at any time. If you had the same GK in City,Pool etc his price would be higher, naturally, but for the game to recognize burnley as a true contender to city,pool etc takes a long time, the game doesnt seem made for instant fast success, its meant for you to take a loooong time to become a genuine PL contender and not just this seasons Leicester.