To my knowledge there is none "close". The Eiffel tower is set in a big park surrounded by historical buildings on one side and the Seine on the other and by a very affluent neighbourhood on the other two sides. You'd have to walk at least a mile before finding a Monoprix, but behind some of the expensive appartment buildings there are many small shops where you'll find almost everything you'd find in a Monoprix and not automatically more expensive.
Good luck in finding what you need.