After an impressive year on loan at Torquay in the National League last season, the 20-year-old has now quickly established himself in the Vale first team under new boss Bruno Ribeiro.
"He's being scouted by Championship clubs," Smurthwaite told BBC Stoke.
"You could see he's come back a man from being at Torquay for a year."
Smurthwaite says that Vale youth product Smith will be offered a new deal by the League One club in the next few weeks, in preparation for what he expects to be firmed-up interest when the transfer window reopens on 1 January.
"I anticipate there'll be some interest in the window, not just at the end of the season," added Smurthwaite. "But it's a long time between now and January.
"There will be a new contract offered to him, as he's on a boy's wage compared to the rest of the squad.
"He's obviously making himself known. And we've been having dialogue with his agent ever since he came in and played his first game in the first team."
Vale stand ninth in League One, on the back of their run of three clean sheets with which they started the new campaign.
Smith has also weighed in with three goals - two in the league and the other in the 1-0 EFL Trophy win over Derby County Under-23s.
Port Vale chairman Norman Smurthwaite is anticipating "interest" in young defender Nathan Smith following his outstanding start to the season.