TLDR: Yes. The USA constitution says the government has power to borrow and lend and to erect buildings where hired people do that. If they combine those powers to erect a building where hired people borrow and lend money, that’s a bank, one that operates on behalf of the national government. Therefore the government has power to create and operate a national bank.
Now here is the longer version.
Background and motivation for researching this
In the USA the government claims to follow a foundational legal document called the constitution. All laws created by the USA government are supposed to be compatible with this constitution. If a USA citizen thinks that one of their country's laws is not "constitutional," they can try to convince a federal judge to publish a ruling that the law is "unconstitutional." If that happens, it means the law is invalid and cannot be enforced by any government agent such as a policeman.
In my experience, government agents actually abide by the decisions of these federal judges. I'm actually not sure why they value the opinion of federal judges so highly, but they really do seem to, at least in my experience. As a result, it can be useful to come up with a convincing argument for why you think a law is unconstitutional, because if you can get a federal judge to agree with you, you might get a law "canceled" -- which is nice because I like the idea of having fewer laws. Also, I just find this stuff personally interesting. So sometimes I spend a bit of my free time analyzing government laws to check if I can make a convincing argument for why they are unconstitutional.
Today I decided to check out the constitutionality of a national bank. This topic appeals to me as a bitcoiner because bitcoin is an alternative to banking and our current banking system is confusing and seems broken, so maybe with a bit of research I could help get it canceled. (Answer: nope, didn't work -- my research convinced me that a national bank is, unfortunately, justified by the text of the constitution.) Here are some of the things I found, just in case anyone is interested.
Provisions of Article 1 section 8
Article 1 section 8 of the USA constitution says the government has these powers, among others: "To borrow Money on the credit of the United States;" "to pay [its] Debts," "To coin Money…[and] regulate the Value thereof." It also gives them power “for the Erection of…needful Buildings,” and “To lay and collect…Imposts” – i.e. fees. It implies that they may issue securities (which include bonds) when it gives them this power: "To provide for the Punishment of counterfeiting the Securities and current Coin of the United States." source
Article 1 Section 9 says the government has power to make expenditures from a treasury and report income from prior expenditures: "No Money shall be drawn from the Treasury, but in Consequence of Appropriations made by Law; and a regular Statement and Account of the Receipts and Expenditures of all public Money shall be published from time to time." Article 2 section 2 gives the government power to appoint people to carry out government functions: “[The president] shall have Power…[to] appoint…public Ministers…and all other Officers of the United States.” Article 2 Section 3 says these officers may be paid, which means the government may hire people: “[The president] shall Commission all the Officers of the United States.” source
Application to the borrowing power
The power to borrow money and hold it in a treasury is explicit in the USA constitution. But borrowing money and holding it in a treasury is the same thing as operating a fund that people can put money into and take it out of later. That’s a bank. The government also has explicit power to charge fees for this service, because it has a discretionary power to lay and collect “imposts,” which are fees. The government also has power to pay an interest rate on borrowed money: “The Congress shall have Power…to pay [its] Debts.”
That’s not the only place that gives them power to pay interest rates, it’s also implied in their power to pass laws to spend the treasury’s money (see article 1 section 9, quoted above). That power is a discretionary power to spend whatever money is in the treasury as long as they pass a law saying what they will spend it on. If they pass a law that says they will pay a certain interest rate on money that they borrow, then voila, they have a place where people can deposit money and get it back later with interest – i.e. they have a national bank.
Application to lending power
Now any bank that chooses to pay interest to its depositors must make an income somehow with the money they borrow. They usually do this by lending the money out at a higher interest rate than the one they pay to depositors. The government doesn’t necessarily have to operate this way but it is empowered to, because the power to loan money is, I think, implicit in the aforementioned powers. Loans are a type of expenditure, and the government is explicitly allowed to make whatever expenditures they want as long as they pass a law authorizing the expense.
For example, when the government buys a bond instead of issuing one, they effectively loan money out and expect it back later – usually plus interest. The government is explicitly authorized to make discretionary expenditures, and loans are just one of those – a purchase made now of something to be received later (i.e. a reimbursement plus interest).
Therefore the government is authorized to operate not just a bank, but a pretty standard one, with the ordinary powers of borrowing and lending for profit just as you’d expect a bank to do. As a bitcoiner, I don't really like this conclusion, but there it is. At least I had fun researching it.