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Once a bank has agreed to make a loan, it then borrows the same amount of money in the interbank market at a slightly lower rate. The lending comes first, the borrowing to fund the loan comes afterwards. This is why so many loans are pegged to LIBOR: Banks charge borrowers rates that are set to levels at some point above what the banks themselves pay to borrow.
A very similar misconception applies when the government spends and borrows. People imagine that the government must first collect taxes or borrow money in order to have funds to spend. In reality, the government just spends what it wants, and then collects taxes in order to balance out the effect the spending has had on the money supply.
In short, banks lend first, fund later. Governments spend first, fund later.