Meaning of “threaten”

Question

According to Merriam-Webster, threaten has the following four meanings:

1: to utter threats against

2a: to give signs or warning of

2b: to hang over dangerously

3: to announce as intended or possible

4: to cause to feel insecure or anxious

There is one sentence:

The lecturer's monotonous delivery threatened to put us to sleep.

What does threaten mean in the above sentence?

Answer 1

A different dictionary gives better definitions here. It's always worth consulting more than one dictionary if the first is inconclusive.

threaten
verb
[with object]
2 cause (someone or something) to be vulnerable or at risk; endanger:
    a broken finger threatened his career
    one of four London hospitals threatened with closure
        • [with infinitive] seem likely to produce an unpleasant or unwelcome result:
          the dispute threatened to spread to other cities

In threatening to put an audience to sleep, it's the "seem likely to produce an unpleasant or unwelcome result" meaning:

The lecturer's monotonous delivery seemed likely to produce the unwelcome result of putting us to sleep.

Answer 2

From your list

to give signs or warning of

Fits. It was so boring, that it gave signs that it could put the listeners asleep.