What 74xx series are you talking about?
74HC03, 7403 (actual), 74S03, 74LS03?
What is the current requirement of the relay, or the coil resistance?
ULN2803's should be able to be driven directly by TTL.
However, the maximum practical sink current is about 350mA; above that, Vce gets to be rather high, and the IC will heat up.
Come to think of it, if it satisfies your logic state, and if it's a low current relay, the 7403 can drive (Current Sink) a relay directly when the output is in the logic Lo state.
Sometimes I get the feeling that AAC gets stuck somewhere in the 'Space/Time Continuum'. I really don't like parroting other members as done here, but when I posted 'both' of my posts no one had replied as yet. Now my posts are last in the list.