The 24-year-old paceman was withdrawn from Pakistan's one-day squad hours before the team's departure on Nov 1 when his injury became aggravated.
"I hope to play a part in the Test series against India because I am feeling better. Once I start bowling, a timeframe can be set for my return," Asif told AFP.
Pakistan face India in the fourth one-day international at Gwalior on Thursday. India lead the five-match series 2-1.
The three-Test series starts in New Delhi from Nov 22. The other two Tests will be played at Calcutta (Nov 30-Dec 4) and at Bangalore (Dec 8-12).
Pakistan selectors are due to name the Test squad on Thursday but Asif's name will not be considered.
"We will only consider Asif's name once we get a clean sheet on him from the medical board," chief selector Salahuddin Ahmed said.
Asif initially suffered the injury to his bowling arm ahead of Pakistan's first Test against England at Lord's last year.
It recurred during Pakistan's home series against South Africa earlier this month. Asif missed the first three one-day internationals before selectors risked him in the fourth one-dayer at Multan where he felt pain while bowling.
The Pakistan Cricket Board last week said the paceman would be sent to Australia for further assessment of his injury.
"This is a different sort of injury. I can lift weight and have started throwing the ball but will have to wait to bowl," said Asif, who first played for Pakistan in 2004.
He failed to take a wicket in his first match against Australia at Sydney but returned to the Pakistan team after taking ten wickets in a side match against touring England the following year.
Asif has taken 51 wickets in 11 Tests and also has 31 wickets in 29 one-day internationals.
Regarded as one of the fast rising pacemen at international level, Asif's career has been dogged by injury problems and a doping scandal.
He and fellow paceman Shoaib Akhtar tested positive for the banned steroid nandrolone in October last year. Asif was banned for one year and Akhtar for two years.
The bans were later lifted on appeal.