Why does a decrease in interest rate reduces the velocity of money circulation? I have seen sources that claim high interest rates increase the velocity of money circulation, but haven't really seen any concrete explanation for this. This is how I think it works: High interest rates -> high opportunity cost of holding money -> Demand for money decreases -> people dont want to hold money and will spend it -> higher velocity of income.

On the other hand,wouldn't a high interest rate make saving more attractive and thus reducing the velocity of income circulation?

EDIT: Sources that apparently claim this are referred to below:

This source does not explicitly state the relationship but somewhat explains the causes: http://thismatter.com/money/banking/money-demand-money-velocity.htm

A wikipedia page that seems reasonably credible states this relationship, but I am unable to fully understand their explanation:
https://en.wikipedia.org/wiki/Velocity_of_money

This exam question from Cambridge International Examinations that explicitly states the relationship (the correct answer is D)
You are absolutely right. Since higher interest rates increase the opportunity cost, you wan't to get rid of it faster (at the margin).

The core of the argument is that the cost stems not only from the amount of money holdings, but also from the duration. Say you are constrained about the amount of money holdings you need in your daily life, but you can still make sure the money is not idle for too long. 

For the second part of your question, you are mixing up concepts. The interest rate influences the Consumption/Savings Decision, but it is the same decision problem as in a (model) world without money, i.e. it is a choice between real variables, whereas money velocity is a monetary question.