The bull market is only now approaching "middle age," and could last up to 20 more years, closely followed strategist Tom Lee told CNBC on Thursday.

"It could last to 2035" or "2038," the Fundstrat Global Advisors co-founder said on "Squawk Box." "That's going to coincide with millennials peaking."

The market is "already growing a little mustache, so it's not a teenager anymore," he said.

While it can be a bit of a moving target, the Pew Research Center defines the millennial generation as those born between 1981 and 1996. The oldest millennials are moving into their upper 30s, which tend to be the start of higher earning years in their careers.

In a note to clients Wednesday, Lee said the stock market had finally reached a bottom after a tumultuous October, and predicted equities would rally about 10 percent into year-end.

Markets suffered sharp losses in October, under pressure from the U.S.-China trade dispute, concerns about the Federal Reserve's seemingly more aggressive path higher for interest rates, and worries about corporate earnings growth.

November begins a traditionally bullish time of year for stocks: The Dow Jones Industrial Average, , and Nasdaq have all posted gains in each of the past five years. The last time all three registered November losses at the same time was in 2010.

In the CNBC interview, Lee said he expects the market to rally at least 13 percent over the next three months.

"We believe the sell off is behind us," said Lee, former J.P. Morgan chief equity strategist. He urged investors to buy stocks hit the hardest by the October sell-off, saying they will "bounce the most."

The current bull market rally, which started March 9, 2009, became the longest one on record since World War II in August, avoiding a 20 percent or more decline, according to S&P Dow Jones Indices.

—CNBC's Kate Rooney and Peter Schacknow contributed to this report.