You can't use прошу when you are asking for something. It's more like "you're welcome" (response to thank you). Прошу is also commonly used when you want someone to repeat what they said (because you didn't hear it the first time)
If you use it as a question ("Прошу?") it means "Pardon me?" (when you didn't quite hear what the speaker said and want a repetition).
Literally it means "(I) ask" or "(I am) asking". The whole sentence "Я прошу тебе" might substitute "будь ласка" if one wants, but that would sound a bit needy and even desperate.
Only if you use it as an answer to "Дякую" you can translate it to "You're welcome". Not when you just say "You are welcome here", definitely no.