Most mainstream auto lenders won't. Here's who does, and what to do instead.
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An ordinary car loan is secured against the car. If you stop paying, the lender repossesses it and sells it to recover the money. That arrangement depends on the lender being able to predict what the car will fetch. A rebuilt title car is hard to value, slower to resell, and its condition depends on repair work the lender can't inspect. The collateral is weak, so the loan is a worse deal for them than the same amount against a clean-title car. It's rarely a judgement about you. Plenty of buyers with strong credit are declined on the car, not the borrower.
Credit unions are the most common yes. They lend to members rather than to a rigid national policy, and many will consider a rebuilt car -- particularly if you already bank with them, the car is newer, and the loan is a modest amount. Small local banks work similarly. Some specialist and subprime auto lenders will also do it, though usually at rates that deserve careful arithmetic before you sign. Buy-here-pay-here dealers finance almost anything, and are generally the most expensive money in the market. Treat them as a last resort.
Because the problem is the collateral, the common workaround is to borrow without using the car as collateral at all. A personal loan from a bank or credit union is assessed on your income and credit rather than the vehicle, so the title brand simply doesn't enter into it. Rates are typically higher than a secured car loan and lower than subprime auto finance. The trade-off is real but simple: you'll usually pay a bit more in interest, and in exchange the lender has no say in which car you buy.
If you're going to ask a credit union or local bank, make the car easy to say yes to. Bring the rebuilt inspection certificate, the repair documentation, and photos. If you can, bring an independent appraisal -- it answers the lender's actual question, which is what the car is worth if they end up owning it. Apply before you commit to a purchase. Knowing what you can borrow, and on what terms, is worth more than finding the perfect car and then discovering nobody will lend against it.
Worth saying plainly: the reason many people look at rebuilt cars is to spend less, and financing works against that. A rebuilt car often costs a fraction of the clean-title equivalent, which puts it inside cash range for buyers who'd otherwise be taking on a loan for a clean-title car. Paying cash also removes the insurance problem -- no lender means nobody requiring full coverage on a car where full coverage is awkward to buy. If the choice is a financed clean-title car or a cash rebuilt one, run both numbers over the whole term rather than comparing sticker prices.
Most large national auto lenders won't, because the car is weak collateral -- hard to value and slow to resell. Credit unions and small local banks are much more likely to consider it, especially for members with an existing relationship. It's worth asking several rather than assuming.