Manager Brendan Rodgers has confirmed that Liverpool are in talks with forward Daniel Sturridge over a new contract.
Karl Darlow is hoping to repay the support he has received since agreeing a move away from Nottingham Forest.
Reports suggested his future would be discussed by the club's directors following the midweek defeat at St Johnstone.
The 1-0 reverse saw Dundee head into the winter break nine points adrift of nearest rivals Ross County.
However, having been promoted to the Clydesdale Bank Premier League as late replacements for Rangers, Dundee chiefs have acknowledged the difficult task faced by Smith this season.
A statement on the club's website, following a scheduled board meeting on Friday, read: "Media speculation over the last few days has caused great uncertainty at the club and this has been very unfair to the manager, the fans and to everyone involved at Dundee FC.
"The manager, the CEO and the board of directors are confident we can still put on a good show in this division and that we can be in the mix to the very end.
"We hope you will continue to stand with the manager and the team as we endeavour to fight our way out of the situation we find ourselves in.
"In the meantime, CEO Scot Gardiner and the board of directors continue to seek out new investment in order to build a secure future for Dundee FC.
"Once again, it is time for everyone at Dens Park to stand up and be counted and we hope you can stand with us.
"The board of directors would like to thank the loyal Dundee supporters for their fantastic backing of the club this season.
"This is a season that was always going to be difficult considering the extremely unusual circumstances that led to us being elevated into the SPL only one week before the league started.
"While the season has obviously not gone as well as we would have hoped for on the field, and our injury list has been horrendous, there is no magic wand that can be waved to fix everything.
"We expect that the manager and the team will continue working hard in training and on match days to climb up the league.
"But, in all reality, we were always going to find it tough against the established SPL teams and those who knew what they were preparing for in the summer."
Dundee's SPL survival bid could be boosted by the arrival of new players this month.
The statement continued: "As a board, we will redouble our efforts with the manager to try to add more quality to the squad.
"Indeed, this has continued during December, and over the last few days, when both the CEO and the manager have been working on targets to bring into Dens for the second half of the season.
"We hope to see the fruits of this hard work at sometime in January."