I was in the same boat as you - DD saying the same things, memories of my own childhood etc. I am reading "How to make sense of your child's friendships" by Elizabeth Hartley-Brewer which I've found very reassuring. DD has found her feet now.
I'd echo the suggestion of a play date or two with children from nursery to build confidence - and also suggest to DD that she thinks of games herself and asks who wants to play that - so "would you like to play hide and seek?" rather than "will you play with me"
Interesting, my DD (3y 2months) has been saying it a couple of times during the last few weeks, and I thought she picked it up from a TV. Mine also goes to a nursery but she does not have a best friend as such.
I was a bit worried though as I am not very good at socialising either and I always looked out for her. So, I decided to show her how to make friends. I have taken her to a park and showed her how to pick a girl around the same age playing alone, and go and ask her name, and introduce herself. "Hi, what is your name?" "my name is Z" " Do you want to play with me?" (and hold hands?) And off they go!
She made few friends this way, some are hit and miss, had few blank stares, but mostly successful and I can see with few practices she feels more comfortable doing this now on her own. I give her loads of positive comments afterwards saying it is nice to meet new friends, and she has been brave etc. I think she is secretly pleased with herself. Hope it helps!
I haven't any great advice but really feel for you - that must be tough for you both.
I think that as it's obviously troubling her, you could talk to one of the staff about how they can improve her interaction and if there are particular times of the day she seems to find more difficult.
And are there one or two children that might be close to her in age that you could try to get her together with outside nursery? Especially with the summer hols coming up, some mums might be glad of a playmate or two to fill time. My DD1 who's 3.3 started a new nursery after her birthday and is still not mixing brilliantly but adores the main teacher/carer and so seems to be coping, but as a shy child myself, I hope someone posts some great ideas for you!
3.5 yo DD has been telling me for a while that she doesn't want to go to nursery and that the other kids don't want to play with her. I don't know how seriously to take this. I've asked staff a couple if times and they acknowledge that she is shy, but is starting to come out of her shell. She is DD number 2 of 3, and potentially used to her old sister looking out for her. I don't know what else to tell you really, other than I always feel a bit sick when she says it.