<PAGE>   1
 
                                                                      EXHIBIT 24
 
                               POWER OF ATTORNEY
 
     KNOW ALL MEN BY THESE PRESENTS that each of the undersigned, an officer or
director, or both, of AVCO FINANCIAL SERVICES, INC., a Delaware corporation,
does hereby constitute and appoint HERBERT F. SMITH or LAILA B. SOARES with full
power of substitution to said attorney, as the true and lawful attorney and
agent of the undersigned, to do any and all acts and things and to execute any
and all instruments which said attorney and agent deems advisable, of AVCO
FINANCIAL SERVICES, INC. to comply with the Securities Act of 1934, as amended,
and any requirements of the Securities and Exchange Commission with respect
thereto in connection with the filing under the Securities Act of 1934 of an
Annual Report pursuant to Section 13 or 15(d) of the Securities Exchange Act of
1934 on Form 10-K for the 1997 year of AVCO FINANCIAL SERVICES, INC., as well as
any and all amendments to said Report, including specifically, but without
limitation of the authority hereby granted, the power and authority to sign his
or her name as an officer or director, or both, of AVCO FINANCIAL SERVICES,
INC., as indicated opposite his or her signature below, to said Report, and any
such amendments, and each of the undersigned does fully ratify and confirm all
that said attorney, or any of them, or the substitute of any of them, shall do
or cause to be done by virtue hereof.
 
     IN WITNESS WHEREOF, each of the undersigned has subscribed these presents
this 25th day of February, 1998.
 
<TABLE>
<CAPTION>
                      SIGNATURE                                            TITLE
                      ---------                                            -----
<C>                                                    <S>
 
                   /s/ N. R. ATON                      Director
-----------------------------------------------------
                     N. R. Aton
 
                    /s/ R. BUKOW                       Director
-----------------------------------------------------
                      R. Bukow
 
                 /s/ L. B. CAMPBELL                    Director
-----------------------------------------------------
                   L. B. Campbell
 
                   /s/ S. J. DAVIS                     Director
-----------------------------------------------------
                     S. J. Davis
 
                   /s/ G. L. FITE                      Director
-----------------------------------------------------
                     G. L. Fite
 
                 /s/ S. A. GILIOTTI                    Director
-----------------------------------------------------
                   S. A. Giliotti
 
                 /s/ J. F. HARDYMON                    Director
-----------------------------------------------------
                   J. F. Hardymon
 
                  /s/ W. W. JUCHATZ                    Director
-----------------------------------------------------
                    W. W. Juchatz
 
                    /s/ S. L. KEY                      Director
-----------------------------------------------------
                      S. L. Key
 
                   /s/ W. R. LYONS                     Director
-----------------------------------------------------
                     W. R. Lyons
</TABLE>
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<TABLE>
<CAPTION>
                      SIGNATURE                                            TITLE
                      ---------                                            -----
<C>                                                    <S>
 
                  /s/ W. J. PEARSON                    Director
-----------------------------------------------------
                    W. J. Pearson
 
                /s/ E. R. SCHUTT, JR.                  Director
-----------------------------------------------------
                  E. R. Schutt, Jr.
 
                   /s/ H. F. SMITH                     Director
-----------------------------------------------------
                     H. F. Smith
 
                  /s/ J. C. SPENCE                     Director
-----------------------------------------------------
                    J. C. Spence
</TABLE>