Ireland were eliminated from Euro 2024 qualifying after managing to score six points from eight matches, with the two wins coming against lowly-placed Gibraltar. They ended their qualification campaign with a 1-0 defeat away to the Netherlands on Sunday.
The draw in the friendly match against New Zealand, ranked 103rd in the world, at Dublin is expected to be the last competitive outing for Kenny.
"Obviously the board are meeting next week. They've a decision to make and I respect whatever that decision is," Kenny said on Tuesday.
"Of course, it would be a dream to carry on and manage the team, but my instinct is that's not going to happen. The evidence suggests that probably won't happen, so I respect that as well."
The 52-year-old replaced Mick McCarthy in 2020 and signed a new contract to remain in charge until after Euro 2024.
"It was an emotional dressing room with the players there. 90% of the players, maybe over 95% of players, their careers are on an upward trajectory and they're only going to improve as players and as individuals," Kenny said.
"It's been a privilege in that regard, the greatest honour you can have. Whatever you did in life, it would be a step-down, no matter what you did, but that's the way it is.
"We have had a lot of setbacks and I suppose that's why I'm not getting a new contract if that's the case. International football is ruthless, that's the nature of it. I know that, I understood that, but that's the way it is."
