The Times has reported that Evans, who in 2004 offered between £700m and £800m, is willing to offer between £550m and £600m this time.
Trinity Mirror is considering selling off its national newspaper titles as an option in a strategic review announced in August and due to be concluded in December.
The group's share price is down from around 600p a year ago to around 500p this week, having gone below 450p in mid-summer.
According to The Times, Evans has registered an interest with NM Rothschild, the bankers advising the national and regional paper group on its review. It said he believes the Daily and Sunday Mirror form an attractive business, which would be better off in private hands.
The price mooted by Evans may not be high enough to tempt Trinity Mirror and it is unclear whether he also wants the rest of its national newspapers -- The People and Scottish titles the Daily and Sunday Record.
Evans owns the conference, training and sports hospitality company named after him.
The Daily Mirror's six-month average circulation is down 6.15% over a year to 1,637,673, compared with a 3.01% drop in the overall daily market. Its sister title is down 5.29% to 1,440,197, compared with a 4.11% drop in the Sunday market.
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