"might bore" is correct and usable in written English.
You can use it when you want to express the possibility that something could be boring. For example: "This movie looks like it might bore us."
I had to laugh when Bill, with his big presentation looming in this finale, was suddenly afraid he might bore his audience.
Both give shorter shrift to political issues that might bore 21st-century readers (Johnson's Jacobite sympathies) or repel them (his attacks on republicanism and religious dissent).
The mentor benefits from an influx of young energy and vigour and additionally gets to talk ceaselessly about his work in a way that might bore to tears even the most devoted of spouses.
One enthusiasts screaming sport bike may jar the wrists and back of another, and a plush cruiser might bore someone obsessed with speed.
Personally, I've loved Vivaldi for years but the same music might bore you senseless.
This might bore Howard Kurtz, but in the theatre of influence and strategy, it is high entertainment and these are only the opening acts to the 2016 drama.
Solzhenitsyn might bear a grudge, but Amis?
"Did you see that?" he asked his daughters and anyone else who might bear witness.
The Piggyback Yard, in its new incarnation, might bear the slyly altered name Piggy Backyard.
Wealthier retirees might bear a larger share of their medical expenses.
It's unclear who might bear the cost of providing the additional capital.
