Pfizer has identified a lead drug to treat the coronavirus, the company's chief scientific officer, Dr. Mikael Dolsten, said Thursday.

The possible antiviral, "if successful, would be given early in the disease process, when patients are admitted to the hospital," Dolsten told CNBC's Meg Tirrell.

Dolsten said the potential treatment has so far shown positive signs in preclinical work. The drug seeks to "counter the expansion of the virus," Dolsten said on "Power Lunch."

Pfizer had initially planned to enter clinical trials for a COVID-19 drug by the end of the year, according to Dolsten. Now, it hopes to start those trials in the third quarter, meaning "just a few months from today," he said.

Pfizer also is working on the development of a COVID-19 vaccine alongside German firm BioNTech. The two companies, which announced their partnership last month, remain on track enter human trials by the end of April, Dolsten said.

He said the companies now expect — if the vaccine is successful and regulatory approval is granted — to be able to "supply millions of vaccine doses by the end of 2020."

"This is much faster than the original prediction of 18 months," Dolsten said. "It's almost half the time."

Shares of Pfizer closed up 2.3% on Thursday at $35.39 after The Wall Street Journal first reported the news of a potential promising COVID-19 treatment. The stock has climbed more than 25% from its March 23 low.

The S&P 500 rose 1.4% on Thursday.