The West Country outfit are firmly in the promotion picture in League One and it would appear manager Gary Johnson has been given the green light to bolster his squad in the January transfer window.
Winters is keen to return to Britain and he has headed to Ashton Gate, having had a trial at Oldham recently.
"Robbie will be on trial with Bristol City for one week and Oldham have asked me to keep them updated," Winters' agent Phil McTaggart told Bergens Tidende.
"He wants to leave Bergen. Should he get a good offer from Bristol City, he will sign."
The striker was pleased with his trial at Oldham.
"The manager knows what I can do and I feel there is a good squad," said Winters.
Sky Sports