Northampton manager Chris Wilder has confirmed interest from Premier League and Championship clubs for Ivan Toney.
Notts County full-back Jordan Cranston has left the club by mutual consent.
Law, the younger brother of former Fir Park midfielder Nicky, now at Rangers, knocked back offers from England's League 1 and League 2 to sign for the Steelemen after quitting Skrill Premier outfit Alfreton Town.
The Nottingham-born player spent some of his youth career with Sheffield United and Chesterfield before signing as a professional with the Spireites.
Law told MFC TV: "I am really delighted to be signing for Motherwell and I can't wait to get started.
"Nicky obviously enjoyed two great years at the club and couldn't speak highly enough of the place, the people and the fans.
"I've been about for games and training so I think I've got a good feel for it. It's obviously a step-up for me from my previous club, but it's a challenge I am relishing."
Well boss Stuart McCall added: "We've had Josh up for training several times last season to look at him and we think he'll really add something to the squad.
"He's a similar type of player to his brother in that he's a midfielder who likes to get on the ball and create.
"We are really pleased to have him here and I am sure he'll add something a bit different to what we have currently."