The Securities and Exchange Commission could approve the funds, known as spot bitcoin ETFs, as soon as next week. Approval would mark a watershed moment for the industry, allowing investors to purchase bitcoin in their brokerage accounts as easily as stocks. (Funds that track bitcoin futures are already on the market.)

A dozen or so asset managers, including traditional finance players such as BlackRock and Fidelity Investments, have joined the race to launch the first spot bitcoin funds.

Excitement about the ETFs drove the price of bitcoin above $45,000 in recent days, its highest level in nearly two years. The token more than doubled last year in a rally that was turbocharged in June when BlackRock—the world’s largest money manager, with a near-perfect record of ETF approvals—threw its hat into the ring. Bitcoin is currently hovering around $44,000.

Bitcoin bears caution that regulatory approval is no sure thing, pointing to the SEC’s rejection of such funds going back to 2017. The agency has long argued that they would be vulnerable to fraud and market manipulation. And even if the funds are approved, some strategists say they expect the green light to be a “sell the news” event for bitcoin because it has run up so far so fast.

“People who bought bitcoin at low prices are sitting on really high unrealized profits,” said Julio Moreno, head of research at the data provider CryptoQuant. “Historically, when that happens, the price tends to correct.”

He predicted that bitcoin could fall to $32,000 if traders take advantage of the ETF launch to take some profits off the table.

The SEC faces a final deadline Wednesday to approve or reject the first applicant: a fund run by a joint venture of Cathie Wood’s ARK Investment Management and the crypto asset manager 21Shares. The agency previously deferred that decision three times. If the SEC changes its tune, it is expected to approve similar applications from other asset managers at the same time to avoid any perceived favoritism.

“It is completely unprecedented because historically we have not seen a situation where multiple issuers are going to be lined up to launch a high-demand product on the same day,” said Nate Geraci, president of ETF Store, an investment-advisory firm.

A fee war is expected to break out in the crowded market if the funds are approved. That could turn the race into a money-losing effort for some of the asset managers, at least in the short term. Some of them are taking a page from traditional finance to compete on cost, while others have begun a marketing push.

Invesco, which filed for a spot bitcoin ETF with Mike Novogratz’s Galaxy Digital, has said it intends to waive its 0.59% fee on the fund’s first $5 billion in assets in the first six months. Fidelity’s filing lists a management fee of 0.39%, the lowest among the group so far. The fund from the crypto asset manager Valkyrie would carry a slightly higher fee at 0.8%.

The wild card in the race is Grayscale Bitcoin Trust, which intends to convert to a spot bitcoin ETF upon SEC approval. The trust, which already has about $27 billion in assets under management, currently charges a 2% management fee.

Grayscale’s chief executive, Michael Sonnenshein, has said the firm will reduce the fee immediately upon conversion. The firm’s ad for its fund, known by its ticker, GBTC, has been spotted at more than a dozen major airports in the U.S.

“It’s going to be brutal particularly for some of the smaller issuers to compete with the likes of BlackRock and Grayscale,” said James Seyffart, an ETF analyst at Bloomberg Intelligence. “But the end investor is going to win here because that’s going to drive fees down.”

Another potential hiccup? Because of the cumbersome operational work involved, not all of the funds are expected to be ready to launch immediately, even if they are approved. Asset managers must open accounts with their custodians, set up data flows and complete other crucial steps ahead of time. Behind the scenes, Coinbase Global is listed as the bitcoin custodian on at least nine of the applications.

The SEC has worked with the asset managers, the New York Stock Exchange, Nasdaq and Cboe in recent months to help them clear some of the hurdles. It asked the asset managers to update their applications to specify the funds’ share-creation and redemption mechanism as well as their arrangements with ETF market makers that help ensure efficient trading and liquidity. Jane Street Capital is the market maker of choice for most of the funds.

Write to Vicky Ge Huang at vicky.huang@wsj.com