The 56-year-old New Zealander lies third going into tomorrow's medal-deciding jumping phase.
Todd, who was named rider of the 20th Century by equestrian's governing body, won the second of his Olympic titles in Seoul 24 years ago.
But no rider in Olympic history has ever won three, and that is the prize that could await him tomorrow.
Todd revealed that he had to cope with loose reins from an early stage of the cross-country course on NZB Campino, but he completed and collected only 0.4 time penalties.
'I didn't have the smoothest ride ever - it felt a bit rough in places,' Todd said. 'By about fence six my reins came completely undone.
'It was a big ask for a horse like that to go that quick round a track like this, and certainly not his ideal track because he is such a big, long-striding horse.
'I knew I just had to keep pushing and asking the questions, and he kept responding. I am absolutely thrilled with the horse.'
Like Todd, 44-year-old Klimke is no stranger to being at the business end of an Olympic competition.
She won Team gold four years ago and was fifth individually, while she has also been on previous German world and European title-winning teams.
She could end up with Team and Individual Eventing gold at Greenwich, a feat that was last achieved by Britain's Richard Meade in Munich 40 years ago.
I knew the course was really challenging, and I am so thrilled with my wonderful horse.Ingrid Klimke
'I knew the course was really challenging, and I am so thrilled with my wonderful horse (FRH Butts Abraxxas)," she said.
'He made it easy for me. Four years ago in Hong Kong he did the same thing, and I am very privileged to have him.'
Algotsson Ostholt matched Klimke's faultless display, as she did in dressage over the weekend.
'The course was much better than I had thought,' she said. 'We were very fluent in the beginning, but later on we lost both front shoes.
'I walked the course six times beforehand, and that really helped me. You need to ride the course like driving a sports car.'
Waiting in the wings tomorrow will be world and European champion Michael Jung, who is a brilliant jumper, and could easily pounce for gold if the top three slip up.