Coutinho, 34, has been without a club since leaving Vasco da Gama in February, citing mental burnout.
"We're very happy that you're here," Neymar said on social media, with a video showing Coutinho alongside Santos players and coaches.
"Our squad and our fans will welcome you with arms wide open."
According to local media, former Brazil international Coutinho needs to pass his medical before Santos can confirm his arrival on a contract that runs until the end of the season.