Mosasaurs are close relatives of the monitor lizards (family Varanidae).
"As early as 1800, the Dutch palaeontologist Adriaan G. Camper recognised the monitor (varanid) lizard affinities of these fossil vertebrates (Camper, 1800; Mulder, 2003), well before they were described as Mosasaurus(Conybeare, 1822). Points of similarity between mosasaurs and snakes were recognised by Edward D. Cope. This inspired him to introduce the order Pythonomorpha (Cope, 1869a)."
Lee (1997) suggests that snakes evolved from a marine mosasaur. The monitor lizards undoubtedly share a common ancestor with the mosasaurs, although their exact relationship remains controversial. There are suggestions that mosasaurs descended from a monitor lizard or from an aigialosaur, which is itself considered a close relative of the monitor lizards. Therefore monitor lizards, snakes and mosasaurs all appear to be closely related. Their exact relationship with each other would need to be worked out.
Yes, they are closely related, but it's much more likely that Mosasaurs either evolved from early Monitor Lizards or that they just share a common Lizard ancestor, since it's unlikely for a fully aquatic Tetrapod with flippers like that to re-evolve limbs fully suitable for land again.