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"Do you have a girl (/ boy) friend?" An alternative is "Are you seeing someone special?".

"Do you have someone you want to go out?" As an alternative to this, do you say this:

"Are you noticing someone special?"

I meant "noticing" to be "atracted by".

This case is that one has never gone on a date with her and wants to do because he dug her.
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"Is there someone you would like to go out with?" As an alternative to this, do you say this:

It doesn't sound right to use 'someone special' about a person that someone doesn't really know. Actually, using 'seeing someone special' sounds rather twee in your first example, it would be more common just to say 'Are you seeing anyone?'.

Alternatives to 'Is there someone you would like to go out with?' are

'Have you seen anyone you fancy?'

'Is there anyone you really like?'
Comments  
Wow! how nice they sound!

Thanks! I really needed that.