1

When I want someone to keep in touch with me, what is the correct one among this following two options?

Please, contact me as soon as you can.

Please, contact with me as soon as you can.

3

"Please contact me as soon as you can" is correct.

In this sentence "contact" is the verb that you're asking the other person to perform.

You can alternatively say "Please keep in contact with me", but you can't really add "as soon as you can" in this case because the sentence implies you're already in contact and that you'd like to remain in contact.

As an aside, your initial question should begin "When I want someone to keep in touch with me...", not "in touching".

1

The first sentence is correct. The second isn't because the verb contact is a transitive verb and must be followed by a noun. The only scenario I can think of in which you would use contact with is when you are talking about being in touch with someone. Examples are

I lost contact with him.

I have stayed in contact with Jane since high school.

In those examples, observe that contact isn't the verb. It is the direct object of the verbs lost and have stayed so in that case it can be followed by with.

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