I agree that if you're worried, it's always worth getting professional feedback and reassurance.

But FWIW, my DD was exactly the same. I work with children with autism (one of the markers of which is lack of eye contact), and by 12 weeks I was beside myself, convinced that she had ASD and killing myself with worry. Bit by bit her eye contact got better, and the day she finally gave me a big, clear social smile... I was at the MCHN and had just told her of my concerns, and then DD just looked at me and BEAMED!!! Little ratbag!

Looking back, I put it down to temperament. She's very shy and self-conscious and still finds eye-contact very confronting. For ages she wouldn't dance or sing if you were looking and clammed up around anyone new. But in no way is she autistic. I think that for socially sensitive children, eye contact can be very intense and tiring. Just think about if you and I met and I stared you straight in the eye the whole time - it'd be really confronting!

I'd suggest doing things that draw attention to you, without too much emphasis on the eye contact. So, singing with hand actions or playing with interesting objects between her face and yours. I'd tend to avoid doing the big facial expressions or strong reactions to eye contact when she gives it, so that, if she is sensitive, she doesn't find it too overwhelming. Just give her a warm, reassuring smile and a 'hello' and let her look away when she needs to. Babies are pretty good at regulating themselves and how much social connection they can handle, as long as we read their cues.

ETA. Just realised that my suggestion contradicts Keta's. (sorry Hun, that was unintentional!). I think either can work, depending on the temperament of the child. I tend to err on the gentler side for really young children as I don't want to 'overstep' their boundaries and push too hard... but yes, some kids just do 'make you work for it' a bit more and respond well to big, bright reinforcement. I'd probably go there when she was a bit older if she wasn't coming to it on her own.