Despite being linked with several members of Tony Mowbray's squad at Parkhead, the Teessiders have yet to strike any deals.
Reports from Glasgow have suggested Boro have tabled a joint bid for Gary Caldwell, Barry Robson and Willo Flood.
But the Championship club have limited transfer funds available and it is understood they have yet to meet Celtic's valuations.
Strachan himself was in Scotland at the weekend to watch the Old Firm derby, and remained there on Monday to carry out some of the groundwork for his winter recruitment drive.
Strachan, who has also been linked with Rangers striker Kris Boyd, had hoped to strengthen his squad before Saturday's Championship clash with Swansea at the Riverside Stadium, although time is now running short.
Former Newcastle central defender Caldwell, 27, is out of contract at the end of the season and the Bhoys are not prepared to meet his demands for a new deal.
Barclays Premier League side Wigan have also been credited with an interest in Caldwell, while Birmingham's former Rangers manager Alex McLeish, who is expected to have significant funds at his disposal following Carson Yeung's takeover at St Andrew's, has been touted as a competitor for Boyd's signature.
Meanwhile, French club Nice have confirmed their interest in striking a deal to take Middlesbrough's 23-year-old midfielder Didier Digard back to his native country.
The club's director general, Patrick Governatori, told Nice-Matin: "We have to recruit two players, a central defender and a midfielder. In any case it's on the right road.
"We are working on two or three cases among which are those of Didier Digard and [Argentinian club San Lorenzo's] Renato Civelli. There are others too.
"We hope to conclude everything shortly. This could be done this week.
"Digard is expensive, but you have to deal with that. We are discussing it. I think in any case that it will be two loans with an option to purchase."
Digard joined Boro in a protracted £4million switch from Paris St Germain under former boss Gareth Southgate in July 2008, but injuries have limited him to just 34 appearances to date.
He is currently on the sidelines once again with a back problem, but appears to be surplus to requirements under Strachan.