The most common and most likely answer to this crossword clue is the 7 letter word ENGROSS
I've seen this crossword clue in The Daily Telegraph
E
N
G
R
O
S
S
More clues leading to the result ENGROSS
- Absorb European margins before tax
- Arrest English rose taking drug on Sunday
- Force loosened Ron’s extremely strong grip
- Absorb new gen supplied by Arctic explorer
- Monopolise congress after its leader goes mad
- Interest in Eastern songs, reworked, Queen covered
- 144 following headless Ben’s arrest.
- Essentially gent's unsophisticated to engage
- Put pen to paper, no pressure, so plunge right in
- Ghastly snores around middle of night capture attention
Buddy explains!

The straight part
We know that "Interest" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case interest = engross.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does ENGROSS mean?
ENGROSS verb- To write (a document) in large, aesthetic, and legible lettering; to make a finalized copy of.
- To buy up wholesale, especially to buy the whole supply of (a commodity etc.).
- To monopolize; to concentrate (something) in the single possession of someone, especially unfairly.
- To completely engage the attention of.
- To thicken; to condense.
- To make gross, thick, or large; to thicken; to increase in bulk or quantity.
- To amass.