bro if u guys are having staring contests u mine as well go over to her and say i win u blinked first. The signals are there fukn go for it before its to late.
Fukn walk up to her man say hey whats ur name? Im cornileous, ive noticed u around the gym. Compliment her or something. Say well it was nice finally talking to u, and if u can keep the convo goin then do, if u cant then say well ill see u around the gym. Then go about ur gym shit, then u see her at the gym again another day say hey how are u doin blah blah, what are u up to tonight? blah blah, u wanna do something tonight or tommorow? then ur good to go. I really wouldnt recommend the immediate date asking out when u first talk to her cuz thats to quick and it will be very awkward if u arnt one to start up a good convo and keep it goin. Just start talkn to her a bit in the gym and get comfortable with her before u ask her out on a date.