O'Neill, 74, addressed the transfer speculation in his pre-match press conference ahead of Celtic's opening Scottish Premiership fixture at home to Dundee on Monday. The veteran manager, who returned to Parkhead last season to lead the Glasgow giants to a domestic double, was also asked about West Ham United's interest in midfielder Arne Engels.
"West Ham, yeah, I think there's definitely truth in that," O'Neill told Sky Sports Scotland. "I think there's been something put in that hasn't remotely made our valuation for a start."
Turning to interest in Johnston, O'Neill confirmed that Moyes had already been in touch to flag Everton's admiration for the Canada international.
"And the same with Alistair Johnston, Everton have been in contact," he said. "I'm not sure whether there's been a bid in there for that one, but they've been in contact saying that they would be. In fairness, David Moyes did call me last week and say that there would be contact. But I don't know how far that has progressed, if it has at all."
O'Neill conceded that interest from elsewhere was to be expected given the form both players have shown, pointing to Celtic's stance on Engels earlier in the year as evidence of the club's determination to resist opportunistic bids.
"But yeah, that would be disappointing obviously. But I suppose it's to be expected, you know, players who have done well," he said. "We turned down a big offer in January from Nottingham Forest, they came in at the very last minute, so we couldn't do anything. Arne was alright with it, you know."
The Celtic boss acknowledged the difficulty of holding on to players who attract interest from bigger leagues, though he insisted retaining his squad remains the priority.
"But it's difficult to stop people if their ambition is to play in a big, big league. West Ham have been relegated but obviously enhance their financial situation, those things are difficult," he said. "Our job is to try and keep them, try and do our best to keep them. If that's not possible, naturally we'll have to try and get replacements in. To replace those players would be a difficult task."
Johnston, 27, has been a regular for Celtic since joining from CF Montreal and impressed for Canada at this summer's World Cup, while Engels has established himself as a key midfield presence under O'Neill following his switch from Augsburg.
Colby Donovan filled in capably at right-back during Johnston's injury lay-off last season and would likely be first in line to deputise should Everton press ahead with a move.