<PAGE>   1
                                                                     EXHIBIT 24


                              POWER OF ATTORNEY

        KNOW ALL MEN BY THESE PRESENTS, that each person whose signature
appears below constitutes and appoints each of Robert H. Lutz, Jr., Robert L.
Adair III and L. Keith Blackwell, his true and lawful attorney-in-fact and
agent, with full power of substitution and resubstitution, for him and in his
name, place and stead, in any and all capacities, acting alone, to sign,
execute and file with the Securities and Exchange Commission an Annual Report
on Form 10-K for the year ended December 31, 1996 for AMRESCO, INC. under the
Securities Exchange Act of 1934, including any amendment or amendments relating
thereto, with all exhibits and any and all documents required to be filed with
respect thereto, granting unto said attorneys-in-fact and agents, and each of
them, full power and authority to do and perform each and every act and thing
requisite and necessary to be done in and about the premises in order to
effectuate the same as fully to all intents and purposes as he might or could
do if personally present, hereby ratifying and confirming all that said
attorneys-in-fact and agents, or any of them, or their or his substitute or
substitutes, may lawfully do or cause to be done.


<TABLE>
<CAPTION>

           SIGNATURE                      TITLE                        DATE
           ---------                      -----                        ----
<S>                              <C>                              <C>
 /s/  ROBERT H. LUTZ, JR.        Chairman of the Board and        March 28, 1997
-------------------------------  Chief Executive Officer
      Robert H. Lutz, Jr.


 /s/ ROBERT L. ADAIR III         Director, President and          March 28, 1997
-------------------------------  Chief Operating Officer
     Robert L. Adair III

 /s/  BARRY L. EDWARDS           Executive Vice President         March 28, 1997
-------------------------------  and Chief Financial Officer
      Barry L. Edwards           (Principal Financial Officer)


 /s/ JAMES P. COTTON, JR.        Director                         March 28, 1997
-------------------------------
     James P. Cotton, Jr.


 /s/  RICHARD L. CRAVEY          Director                         March 28, 1997
-------------------------------
      Richard L. Cravey


 /s/ GERALD E. EICKHOFF          Director                         March 28, 1997
-------------------------------
     Gerald E. Eickhoff
</TABLE>



                                      -1-
<PAGE>   2
<TABLE>
<CAPTION>

<S>                              <C>                              <C>
  /s/ EDWIN A. WAHLEN, JR.       Director                         March 28, 1997
-------------------------------
      Edwin A. Wahlen, Jr.


  /s/ AMY J. JORGENSEN           Director                         March 28, 1997
-------------------------------
      Amy J. Jorgensen


  /s/ JOHN J. McDONOUGH          Director                         March 28, 1997
-------------------------------
      John J. McDonough


  /s/ BRUCE W. SCHNITZER         Director                         March 28, 1997
-------------------------------
      Bruce W. Schnitzer


  /s/ RONALD B. KIRKLAND         Vice President and Chief         March 28, 1997
-------------------------------  Accounting Officer
      Ronald B. Kirkland         (Principal Accounting
                                  Officer)
</TABLE>



                                      -2-