Macquarie will sound out the Exchange's biggest shareholders about a price at which they would be prepared to sell.
The Sydney-based bank will also meet a range of the Exchange's biggest customers - through the London Investment Banking Association and Apcims, the small stockbrokers' lobby group - to soothe concerns that price rises may accompany an Australian takeover.
Macquarie, which has until Thursday to bid, is thought to be reluctant to offer more than 600p a share. An informal approach at 580p was dismissed as "derisory" by the LSE last week.
The City wants more too. LSE shares advanced to 619p by Friday night, and its two biggest investors have paid more than 600p a share recently to top up their holdings.
After spending the weekend picking over the Exchange's financial information and its trading platform, Macquarie and its advisers must decide how much it would be prepared to pay.
The bank has a reputation for driving a hard bargain. It typically invests in infrastructure businesses with steady income such as airports, bridges, toll roads and ferries.
A £1bn debt package has been lined up through Dresdner Kleinwort Wasserstein, the German investment bank, to part finance any offer.
Macquarie is understood to be prepared to ask Clara Furse, the Exchange's chief executive, to stay on. But she would not be offered a stake in the business.