In an interview this week the England international said his motivation was to build a legacy which would see him spoken about long after he had retired.
But Slot was happy for Alexander-Arnold to set his ambitions high, as long as they contributed to the success of the team.
"I definitely like this in a player. I think this is something every player must have if you play at this level, if you don't have this you probably don't play at this level ," said the Dutchman.
"As long as they understand that they need the team to be special as an individual. As long as they put the team first and as a result be special themselves, that is what they all do - including Trent.
Trent Alexander-Arnold has been strongly linked with a move to Real Madrid
"All the top players have this, they want to be special, they want to be the best. If that's not what you have I think it is so hard to put the effort in to stay fit, to be good every three days.
"You always see them during the game and you probably think 'What a life these players have' and it is true, they have a great life, but they have to put a lot of work in to be able to at their best every three days.
"That is not given to everyone, that mentality and discipline, and therefore that's the mentality of a top player and part of that is you need and want to be the best possible player in your position or the best player in the world."
In terms of fitness, Slot said only defenders Ibrahima Konate (knee) and Conor Bradley (hamstring) remain sidelined as they prepare for the start of the busy festive period with [Sunday's trip to Tottenham](/sport/football/tottenham-vs-liverpool-fc-prediction-kick-off-time-tv-live-stream-team-news-latest-h2h-results-odds-premier-league-2024-preview-b1201135.html).
"These are the two that don't train. It is quite difficult because I thought Conor would have been back already so let's wait and see," he added.
Liverpool's run over the Christmas and New Year period is not excessively taxing with an 8pm Boxing Day kick-off at home to Leicester followed by a trip to West Ham three days later, with a week's break before the visit of Manchester United.
"We put a lot of effort in pre-season and made the decision to give them quite a lot of days off and I still think we find the benefits," said Slot of how they were managing the fitness of players.
"In Holland some of the clubs only gave them seven or eight days off and they got injured.
"I have to make compliments for the performance staff. I think it is only two injuries left and a few are coming back, we only miss two on the training ground."