The most common and most likely answer to this crossword clue is the 8 letter word GAZUNDER
I've seen this crossword clue in The Times
G
A
Z
U
N
D
E
R
More clues leading to the result GAZUNDER
Buddy explains!

The straight part
We know that "lower offer" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case lower offer = gazunder.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 GAZUNDER mean?
GAZUNDER verb- To reduce the offer price of a property after agreeing to a higher one (normally just before contracts are exchanged)
- A chamber pot.