The all-time list for first-week sales has Adele's 25 and Oasis's Be Here Now at the top with 800,000 each.
If Sheeran doesn't quite match them, he could go third on that list, above Take That's Progress, which sold 518,601.
Sheeran has said he wants to challenge Adele's album sales.
"Adele is the one person who's sold more records than me in the past 10 years," he told GQ.
"She's the only person I need to sell more records than. That's a big feat because her last album sold 20 million. But if I don't set her as the benchmark then I'm selling myself short."
His album has already exceeded expectations. Before the release, Sheeran's manager Stuart Camp told Music Week he would be "ecstatic" with first-week figures of 350,000.
But he added: "But really I'm thinking, what will we have sold by the end of 2018? It's about the long game."
The final chart and first-week sales figures will be revealed on Friday.
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