The Stormers are in the market for reinforcements after Imad Khan suffered a serious injury against the All Blacks, while veteran Dewaldt Duvenage was injured in his final match before retiring.
The 60-cap Springbok has been in excellent form for the Cheetahs since joining them for the Currie Cup earlier this year, following his departure from Japan's Yokohama Canon Eagles after four seasons.
Dobson is also considering a move for one-cap Springbok Louis Schreuder, who is currently playing for the Boland Cavaliers.
"Loan players are an option, like Faf and Louis Schreuder from Boland. We are looking at that," Dobson told South African newspaper Rapport over the weekend.
De Klerk, a back-to-back Rugby World Cup winner with the Springboks, began his professional career with the Lions before moving to Sale Sharks in 2017 and later establishing himself as one of the world's leading scrumhalves.
Schreuder, who began his career in Durban, made his provincial debut against the Lions in February 2014 before going on to play for the Sharks, Western Province, Bath and Newcastle Falcons.
Dobson also remains interested in Lions scrumhalf Haashim Pead. The former Junior Bok star is contracted in Johannesburg until 2028, but has found his opportunities limited at the Lions, which could potentially open the door to a move in the future.