The stronger side should offer the draw. If your opponent is about to lose on time that makes you the stronger side. It's gentlemanly to offer the draw in that case but of course not necessary.
Draws and when to offer them?
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Dec 3, 2016
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#3
VladimirHerceg91 wrote:
If I have 1 minute left on my clock and my opponent has 6 seconds, am I considered better even if I am way down on material?
Yes. In fact it's so lost some people will resign before it gets that bad for them (having only 6 seconds left).
It was nice of you to offer a draw. If the position was a super boring dead draw then some players will do that as a nice gesture. If it was a normal position though (with at least some winning chances) then draw offers don't really happen. The player who is low on time is just lost.
No, you should not resign haha 
I offered a draw because my opponent had a winning position, and had little time left on their clock.
I didn't want to resign, so I wanted to draw instead, they declined and lost the game on time. Should have I resigned? If I have 1 minute left on my clock and my opponent has 6 seconds, am I considered better even if I am way down on material?