Make eyes at somebody
Definition 1 of 1
Pro Tip 1/3Nonverbal Flirting
Using 'make eyes at somebody' suggests using nonverbal signals like glances to show interest without words. 
At the cafe, she kept making eyes at someone seated nearby.
Pro Tip 2/3Subtle Approach
The expression often implies a subtle or playful way to show attraction, not an overt or aggressive one. 
During the play, they were shyly making eyes at each other.
Pro Tip 3/3Potential Romance
'Make eyes at somebody' can signal the beginning of romantic interest that might develop further. 
The new classmates made eyes at each other, hinting at a budding romance.