In russian we have a word "приласкать" - verb, which we can use to say, "treat with much tenderness". Usually with animals that means they'll start following you around. What is the English equivalent?
English might use different words depending on the details of the situation. For example, my cat likes to curl up and sleep with me, in which case I say:
I like to snuggle/cuddle with my cat.
In other situations I give him treats because he likes them, in which case others might say
I indulge my cat
I spoil my cat
At other times my cat likes to follow me from room to room, or rub himself against me when he wants attention. In which case I might say:
My cat is very needy.
My cat wants love.
All of these may be implied by that single Russian word, but in English they are separate.
If the Russian is specifically physical, meaning, "if you pet this animal it will follow you", I would go with "caress", as it implies physical contact that invites continued contact.
If it is more metaphorical, where you do something that appears kind but is actually not good for the recipient, I would suggest "spoil". However, someone who is spoiled generally demands that the preferential treatment continue, and has little loyalty to the person spoiling them. If the recipient is more made into an (unwanted) follower, perhaps "indulge".