Having inherited an unbalanced squad from his predecessor, Gordon Strachan, Mowbray is hoping to make changes at the turn of the year, but with finances exceptionally tight, the Boro boss will first have to raise funds and reduce the wage bill.
Wheater, who is out of contract at the end of the season, and O'Neil, who cost £5m when he moved to Teesside in 2007, are two of the club's most saleable assets, and both are effectively available to the highest bidder.
In an ideal world, Mowbray would prefer to lose neither, but it looks increasingly unlikely that Wheater, in particular, will remain at the Riverside beyond the end of the year.
I don't wish to lose any of my players, said the Middlesbrough manager. But I'm not going to sit here and say, we won't do this' or we won't do that'. It's a business and we will have to wait and see what happens.
The frustrating thing with David is that he's got so little time left on his contract. Otherwise, I think he's a real rock, a foundation we can build the club from.
I think there are envious eyes watching him. You hope that somebody is watching all your players somewhere along the line because that means you have a good team With David at this time, the preference would be to tie him up on a long-term contract and keep him here for a long time. But we will have to see how that unwinds over the next few months.
Mowbray has already held a number of informal discussions with Wheater, in which he has outlined his desire to keep the 23-year-old on Teesside.
However, Boro's financial position means they are only expected to offer the centrehalf a short-term deal, and they will be unable to match the top-flight wages he is currently earning at the moment.
That opens the door for Premier League clubs to make an alternative offer, with Wheater having previously attracted the attention of both Stoke City boss Tony Pulis and Everton manager David Moyes.
The defender's value has dropped following a difficult first season in the Championship, but Boro can still hope to receive a seven-figure fee for his services. If he leaves as a free agent next summer, however, they will not receive a penny.
I've spoken to him (Wheater) on a number of occasions,
said Mowbray.
These days, though, it's not just about managing a player.
You go back 20 years and it was just a relationship between the manager and the player.
Unfortunately, today, there are lots of other things in the equation that you have to overcome. There are lots of different opinions.
It will come out in the wash over the next few weeks and months, but our wish is that David signs a long-term contract and stays here. The reality, though, is that when you are in the position we are in, bigger fish can come along and court him.
Both Wheater and O'Neil will be in the starting line-up for this afternoon's home game with Millwall, as Mowbray attempts to propel Boro up the Championship table.
The Saltburn-born boss has now been in position for almost a month, and has seen enough to accept there will be no quick fix to the malaise that has sent the Teessiders hurtling out of the Premier League and towards the trap door to League One.
If anything, the task appears to be even more onerous than he imagined when he agreed to take over from Strachan, although his passion and determination to succeed remain undiminished.
Lots of things have surprised me, he admitted. But we are where we are and we will work diligently to get this club pointing in the right direction.
There are restraints, but we will endeavour to get the club going the right way.
When you have been in the Premier League as long as we have, it leaves a legacy.
You've seen other clubs having trouble coming out of that, and we're trying hard to stabilise and stop things getting worse. We have to do that before we can move forward again.
There will not be a rush of signings or players coming in, so we have to work away to make sure we head in the right direction with the right brand of football.
It can be done. You look at Swansea (who beat Boro 1-0 last weekend) and you see a brand of football that has not taken tens of millions of pounds to introduce. But first we have to get to a base and build on that if we can.
Middlesbrough misfit Lee Miller has joined League One side Notts County on loan until the start of January.
Miller, who arrived in a £500,000 switch from Aberdeen under Gordon Strachan, failed to score in his 12 appearances for the Teessiders.
Source: Northern Echo
Source: Northern Echo