I don't think there is a hard and fast rule here. If you both enjoy conversing back and forth, go for it. The worse thing that happens is that she says no thanks or maybe at a later time. I would go with something like "I really enjoy conversing with you. Would you be open to me calling you?" Hope that helps! Good luck.
Honestly I agree that there is no hard and fast rule. It should be something that you can tell based on your communications with her. If you have been communicating for a while then you should be able to tell if she would be receptive to your asking for her number. I do think though (I am a very old fashioned girl) that it should be the guy that makes that the move to take the communication to the next level. So talk to her and learn to feel it out. That is going to help both in dating and in marriage.
If this is the same girl. I just have to say BOY IS SHE SPECIAL. You better be asking her for her phone number at this point. Good luck.
When you feel like talking to her.
If you both seem interested in getting to know each other, I'd say the sooner the better. You don't have to talk every day, but one phone call at least will be helpful. I would think that would be better than wasting time on the computer. If things go well on that call, maybe the next time you can ask her to meet you somewhere for coffee, or a walk (if you live close to each other that is.) If you don't really connect well , then at least you know.
I usually ask upfront as to what method of communication she feels most comfortable with. ( e.g. cell, email, text, cm message ect.) It seems to work. That way you do not lock yourself in and she does not feel pressure to giving out a phone number). BTW skype / yahoo messenger is great for first contact.
If you've been in touch with the same lady, and the messages are flowing smoothly, ask away at the earliest time possible! I had one gentleman ask within days of sending messages back and forth, and I was open to it. Of course, if she's a little nervous about someone having her number, she may ask for yours, instead, but I think that, for the most part, a lady likes to give out her number to let the gentleman call.
This is a very good question, John. I think a guy should ask for her phone number after they have emailed back and forth a few times and he finds he still is interested in getting to know her. All this texting and emailing is void of emotion and a very difficult way to really discern who the other person really is. Phone calls allow both parties to hear the inflection in the other person's voice and be able to get to know the individual a bit better. But, I also think after a few weeks of phone calls they ought to meet one another face to face. What may be a great Internet relationship or a great phone relationship could be a disaster in person relationship and the only way to know if the time your are spending could produce positive results is to progress to an in person meeting. Building a wonderful email and/or phone relationship that will go no where once you meet the person in person may take time away from a relationship with someone with whom you can enjoy a wonderful, meaningful "in person" relationship.
Just my thoughts on the matter. Will look forward to other responses.