The 32-year-old signed for Scunthorpe on a short-term contract in February this year but is now a free agent after leaving Glanford Park at the end of last season.
The former Brentford and Gillingham striker has been working with The Robins for the past week but, as manager Bobby Gould told the club's official website, has been offered terms elsewhere.
He said: "I had a phone call from Robert Taylor yesterday saying that a Football League club had offered him a contract if he went to play 45 minutes for them, and that a non-league club had offered him a contract as well.
"We were not in a position to discuss that and I wanted to see a little bit more of him in training and playing capacities.
"I've told him that if it doesn't work out then he can come back and we will review the situation. But for now he has departed for Chester and that's the kind of thing that will happen over the next few weeks as people sort themselves out."
Taylor began his career as a trainee at Norwich in 1990 and has spent time at Manchester City and Wolves but he is best remembered for his time at Brentford, where he scored 73 goals in 227 appearances.
