Quoting both of the spells, Sleep (PHB p.276) states:
(…) each creature affected by this spell falls unconscious until the spell ends, the sleeper takes damage, or someone uses an action to shake or slap the sleeper awake.
And Entangle (PHB p.238):
Grasping weeds and vines sprout from the ground in a 20-foot square starting from a point within range. For the duration, these plants turn the ground in the area into difficult terrain.
A creature in the area when you cast the spell must succeed on a Strength saving throw or be restrained by the entangling plants until the spell ends. A creature restrained by the plants can use its action to make a Strength check against your spell save DC. On a success, it frees itself
When the spell ends, the conjured plants wilt away.
So by the spell description, Entangle doesn't cause any damage to the affected creatures, which makes me think it wouldn't wake creatures affected by Sleep spell.
Moreover, unconscious creatures automatically fail their Strength Saving Throws (PHB 292), so they get automatically entangled as well.
Do these entangling vines affect the asleep creatures in any way so they wake up?
Best Answer
RAW, they will not awaken.
The sleep spell clearly lists the ways to awaken the sleeper. This is magical slumber, which can occur even in the pitch of mortal combat (in fact usually in the pitch of mortal combat). It is not ordinary sleep. Note that the action to awaken the sleeper is to take damage, or be shaken or slapped. The spell says nothing about loud noises, bright lights or other things that normally awaken sleepers.
The entangle spell has no language in it that describes a level of violence even so mild as a slap or shake. A DM is free to interpret this otherwise (that's their job), but there is no specific language in the rules to suggest that the entangle spell will awaken a magically slept person.