Boss Nigel Pearson revealed earlier this week the Foxes are planning a sensational swoop for Davids.
Agent Robert Geerlings claims the former Holland midfielder is giving the offer careful consideration.
"He has several options but Leicester is very interesting for him. There will be a decision this weekend," Geerlings told reporters.
The 36-year-old, who has 65 caps for his country, has played for Ajax, AC Milan, Juventus, Barcelona, Inter Milan and Tottenham and is available on a free transfer after his second spell at Ajax came to an end in the summer of 2008.