A frequently repeated claim deployed in critiques of loanable funds theory is that private bank money creation removes the constraint on investment being limited by "prior" savings. In a generally good article on loanable funds here, Servaas Storm spends a lot of time discussing the ex nihilo creation of private money.
I think this is highly misleading. Whilst not denying that understanding bank behaviour is important, the savings constraint issue is simply a result of having a monetary exchange economy and has nothing to do with where the money comes from.
First some clarification. It is sometimes suggested that the constraint in question is that saving must take place before investment. To the extent this really does refer to the order of events in time, it is clearly wrong. Saving and investment must always take place simultaneously, by their very definition, regardless of whether we are talking about a barter or monetary economy.
What does matter is the relationship between plans and outcomes, specifically when agents have plans that are inconsistent. In a normal market for some commodity, if planned demand is different from planned supply, the amount actually traded will be the lower of the two. Neither buyer or seller will trade more than they want.
Translated into a loanable funds market, this means that the amount of actual saving would be the lower of planned saving and planned investment. Savers cannot end up saving more than they planned. And this is indeed what we find in a barter economy, where all saving is in the form of commodities.
The difference with a monetary economy is that actual saving is not constrained by planned saving. This is because actual saving must be equal to actual investment and actual investment is not constrained by planned saving.
The easiest way to see this is to think about bank lending and recognise that banks can provide finance to enable new investment without first needing to check the plans of their depositors. Although this is a useful picture, it can lead to the mistaken view that it is private bank money creation that removes the planned saving constraint. This is not correct. What removes that constraint is monetary exchange and that holds even with a fixed, exogenous money supply.
Consider an economy where there is a fixed money supply of $100, all held by households. Households also hold $100 in loans to firms, so $200 in total financial assets. Firms would like to borrow more and invest more, but households do not wish to take on more credit risk.
Now assume that households become less risk averse and wish to change their portfolio to $50 money and $150 loans. Note the important distinction here between saving and lending. Households are planning additional lending, but they are not planning any increase in holding of financial assets (which we can equate to saving here, as we will assume households do not undertake investment expenditure). Although we talk about loanable funds, we don't mean what is actually loaned but what is saved. Here, the planned saving is zero.
However, if the $50 of additional loans to firms is spend on investment then it ends up back in the hands of households again. Household income has risen and they end up still holding $100 of money, even though they planned to only hold $50. Total financial assets has risen to $250. There has been actual saving of $50, unconstrained by planned saving of zero, without any new money being created.
The point here is that what facilitates the change in investment and therefore actual saving, is not a savings decision, but a portfolio decision. The reason bank lending matters is because it is a form of portfolio decision and, indeed, banks play a large part in the overall portfolio decisions. Money creation matters because it changes the portfolio options for households and may therefore influence their portfolio decisions. It is not the magic ingredient that undoes the loanable funds model.
 Part of the reason this whole issue doesn't figure much in more mainstream economics is that there is a tendency to focus on analysing outcomes that are consistent with plans, and less attention is given to the question of what happens when they are not.