On-loan Chelsea striker Romelu Lukaku said after Sunday's victory over Southampton pushed the Toffees into fourth place they had to be considered outsiders to win the Barclays Premier League.
Everton have a chance to consolidate their position in the top four, five points behind leaders Arsenal, at Stoke on New Year's Day and, while some managers would ban all talk of a title charge, Martinez welcomes it.
"When you are a footballer involved in a league and you come off the pitch feeling satisfied with the players around you and the way the team is working it is only natural you have that forward thinking," he said.
"When you are five points from the top halfway through the season why not? I don't think we have fulfilled our potential.
"You should always have big aspirations and dreams and in the same way have a real humble approach to your work, trying to improve on a daily basis, and that is the theme in the squad.
"You need to have a dream and that is really positive. We have enough negativity in football. It is good when you have a real fresh and healthy dream to fight for."
For Everton to replicate what they have done in the first half of the season over the full campaign they need to hold on to key players in the January transfer window.
There is already speculation about Manchester United boss David Moyes returning to his former club with a renewed bid for left-back Leighton Baines after failing to land the England international in the summer.
The Scot was also linked with midfielder Ross Barkley, although suggestions of a ?50million deal prompted nothing more than a joke from Martinez.
"?50million? We wouldn't let anyone out on loan in January!" he said.
"He has a long-term contract and the first half of the season has been a real joy to see him grow and the second half of the season will be even better."
The Everton boss is not worried by what a month embroiled in transfer headlines will do to the 20-year-old, however.
"It is down to his personality and Ross is ready for all that," he added.
"He has had such a long spell away from the pitch (because of injury) and that and being away on loan has made him mature a lot and he doesn't take anything for granted.
"Anything which comes his way he is embracing it, enjoying it and mastering it.
"Any experience which comes his way will help develop him and I have no concerns at all.
"The bigger the challenge the bigger the response."
Martinez is also confident they can hold off big-money bids from other clubs.
"We have done it before so yes, absolutely. There is not even a doubt," he said.
"You can't fight the trend of the game, but what you can dictate is the timing and everything you do should be for the good of the club.
"The stability we have from a financial point of view allows us to make football decisions and this January window is important for us to make sure we kick on and finish stronger than we started."
The match at Stoke takes priority and, although the Potters were dispatched 4-0 at Goodison Park a month ago and have conceded eight in their last two matches, Martinez is wary of the trip to the Britannia Stadium.
"You look at the last two results and it was a little bit of a freak performance against Newcastle (losing 5-0) and that affected them going into Spurs (a 3-1 defeat)," he said.
"They are unbeaten in their last six at home and we know when they play at the Britannia they are an extremely difficult team."
Source : PA
Source: PA