An idiomatic alternative to "familiar with the matter"

I like in the know (The Free dictionary):

having more information about something than most other people: People in the know go there for the best skiing in the east. (Cambridge Dictionary of American Idioms)

Privy to special or secret information, as in Not too many people are in the know about this project. (The American Heritage® Dictionary)

So your example would become:

An agreement will soon be reached by the two big companies according to people in the know.

I have come across the phrase "according to inside sources" often in newspapers and articles.

Usage:

According to inside sources, an agreement will be reached between the two companies soon

Google Ngram result for "according to inside sources"

I'd suggest, in the loop

Part of a group that is kept up-to-date with information about something: knew about the merger because she's in the loop.

American Heritage® Dictionary of the English Language

An agreement will soon be reached by the two big companies according to people in the loop/ in the loop sources.

Ngram

Even without “the matter” (which would probably reduce the hits for “according to people familiar with”) this Ngram puts “according to informed sources” ahead of the phrase that you are trying to replace.