The online editions of the Financial Times and The Wall Street Journal said Sunday that the deal could be announced Monday. They put the price tag around $6.8 billion, or €5.2 billion.
The newspapers cited unnamed people familiar with the talks.
The offer equates to €9.50 per share. A month ago, TNT rejected an "informal" offer by Atlanta-based UPS of €9 per share. TNT shares closed Friday at €9.35.
A UPS spokeswoman said she could not confirm the media reports. TNT did not respond to a request for comment late Sunday European time.