Based upon his current contract, any realistic contract offer from the team would be deemed by player and agent to be an insult. I would not offer him a contract as a player. I would offer him a contract as a team "goodwill" ambassador and/or system wide catching coach. Depending on how he does at that and what interest he has in coaching, he can then be considered as a viable option to be added to Jake Mauer's staff when he ascends to the majors

Yes, I would consider extending him year to year on a hometown deal if he is willing. 

Sadly, you could extend him until he wants to quit, a la David Ortiz and the Red Sox. The DIlemma is, where does he play and where does he bat. Or, is his bat worth the $12-13-15 million a year you would have to offer him not to offend him.

Of course, he would be free to see what the market might bring him, elsewhere.

And, again, who do we have to honestly replace him in 2019. Diaz is still a few years away. Always thought it might be Kepler, but the Twins aren't giving him time at first anymore. Doubt that it would be Vargas or Park, even short-term. So he is better than nothing.

By the way...what did happen to Justin Morneau this year, and would he have been a good fit for the Twins as a DH this season?

I would offer a 2 year 16 to 20 million. Based on obp. That is hos key stat. We need more .360 + obp in the lineup. Add incentives for plate appearance for another 1 to 3 million per season depending on base salary. He will be 36. I suppose a 3 year offer can go on the table.

Joe Mauer's next contract will be to become part of the Twins ground crew. Hopefully for Joe that they offer paid time off for his September illnesses that he seems to get lately...

The Twins will have the leverage this time. He's comfortable playing at home and won't want to leave the wife and kids. We've got talent behind him. They won't make an embarrassing offer, but wouldn't be surprised to see a hometown discount.