Is using "I don't know that" instead of "I don't know if" grammatically correct?

Solution 1:

The string I don't know that is grammatical and common and is used to express doubt. The use of subordinators if or whether after know would suggest a less firm position: he doesn't know or he's not so certain.

• I don't know that there are bargains yet, but there certainly are opportunities.
• I don't know that the President has said that he fully supports the conference of the states
• I don't know that his chances are any better now than they were then

Solution 2:

Yes, it is grammatically correct, albeit informal. It is an idiomatic use of the verb know, meaning I have doubts about somebody's ability to do something. The entry in the Oxford Dictionaries Online reads as follows:

not know that (informal) used to express one’s doubts about one’s ability to do something: I don’t know that I can sum up my meaning on paper.