The LM-curve is completely independent from the loanable funds market. This is because the loanable funds market is entirely determined by the real side of the economy. Therefore, it is linked to the IS curve, not the LM curve. Even more, equilibrium in the loanable funds market for given $Y$ allows you to derive the IS curve, just as the keynesian cross allows you to do.
To see this, consider both relationships in the loanable funds market. The demand side of loanable funds is given by the demand for investment (negatively related to the real interest rate $r$) and the supply side is given by all savings available (positiveley related to $r$). For equilibrium it is required that $S=I$ holds. Which is exactly the equilibrium condition represented by the IS curve: what are the combinations of $r$ and $Y$ for which savings equals investment.
To derive the negative IS-relationship from the loanable funds market, just go through the following reasoning. Suppose that the loanable funds market is in equilibrium. This equilibrium holds for a given $Y$. If $Y$ increases, and the marginal propensity to consume is smaller than unity, the level of savings will increase for each $r$, so that the savings curve shifts to the right. More income does not affect investment, so that a new equilibrium requires a lower $r$. Conclusion: if $Y$ increases, a lower $r$ is needed for $S=I$ to hold. This is the IS-curve.