How can government budget deficit cause the level of interest rates to rise for an economy?

When price increases, interest rate tends to rise. Government budget deficit suggests high Government spending (G) which leads to the rightward shift of AD and hence the corresponding upward pressure on price.

Interest rate is determined by the money demand and money supply, government budget deficit suggests that government is unable to tap into reserves to finance spending. They will have to borrow. This increases the demand for money and thus causing the interest rate to rise.