franchise tag

coaches some help please, if you franchise tag a player who has turned down previous offers can you try to nail down a long term contract straight away?
also will the player want more money because of the tag?
for example player wanted 14 mil for 4 years but franchise tag is 10 mil for 1 year.
if I tag player what are chances of long term contract for slightly more than what he turned down.
Thanks for any help guys

You can re-negotiate the tagged player's deal during next season, which will prolly be in the same $$ area as this season.

as soon as FA starts or one regular seasons begins you think?
Can't really afford ten mill off my cap for long.