I like the second one a little better, but it doesn't really matter. Note that we don't say "give a phone call" in American English; I would say, "please phone me..." or "please call me".

‘Phone’ and ‘give a call’ sound unnatural. Between the two choices: Please call (me) (at) around 10 today. Please call (me) today (at) around 10. the second emphasizes that the call has to be today, and supplies a time. Depending on the context, ‘me’ and ‘today’ could be omitted, and the ‘at’ is optional too. I’d tend to say it.

At least to me, in 2 it sounds like it's really important that the call is "today". Other than this, they both seem perfectly fine.

They are both grammatically correct and natural.