Earlier in the week the Northern Irishman noted that departed midfielder Stuart Armstrong and winger Patrick Roberts had yet to be replaced and his fears that Celtic would lose out on target John McGinn were realised when the Hibernian midfielder signed for Aston Villa on Wednesday.
The Scottish champions have signed striker Odsonne Edouard from Paris St Germain - for a club record fee which is reportedly £9million - and goalkeeper Scott Bain from Dundee, both of whom were on loan at Parkhead last season.
Asked if the board matches his ambitions, Rodgers said: “I wouldn’t answer that. It’s private”.
The former Swansea and Liverpool boss then said: “I have a huge respect for the people here who work at the club.
“The board of directors have created a very stable foundation here for the club to work under for a number of years.
“I have to respect that. Of course as a manager I always want the club to be the very best it can be, for the supporters.
“Inside I am a supporter at heart so I want to see the club progress but I am also the professional manager of the club and I have to understand maybe where the limits are at times.
“But that doesn’t stop me wanting to push to do the very best that I can for the supporters but also for the players, for us to give the players the help that they need.
“They have been incredible here for the period of time I have been here, the consistency they have shown and the quality they have shown.
“I repeat, all we are looking to do is try to replace really, firstly what we lost and two, to try and improve the squad.
“I want Celtic to be the very best that it can be and whatever length of time I am here, I will always push for that.”
Rodgers does expect Australia international Daniel Arzani to soon sign on loan after the 19-year-old playmaker completed his move to Manchester City from partner club Melbourne City on Thursday morning.
He is also hopeful that 32-year-old Honduras international Emilio Izaguirre will return to the club to provide cover for left-back Kieran Tierney.
Of a deal for Arzani, he said: “We are hopeful that it can be. He has completed on the Man City side, virtually most of it is completed on our side.
“He has been up here today so, like I say, we hope for that to complete over the next few days.
“He hasn’t done anything since his last game with Australia a little while back so he is going to have to come in and adjust and get up to speed before we can even consider him for a game”.
Izaguirre is set to return to Celtic a year after leaving the Scottish champions for Saudi Arabian outfit Al-Fayha.
Rodgers said: “It is something that will hopefully be complete in the next 24 hours. Young Kieran Tierney is clearly our number one left-back, he is an outstanding player.
“If we can get Emilio in it will give us that cover and experience.
“When he was here he was exceptional as a professional. He is loved by the other players, he trains every day, he is a role model as a professional and like I say when he was asked to play, he comes in at a really good level.
“If we can do it, he would be a really good addition to our squad to cover Kieran.”