Marie Husbands, 74, was admitted to hospital when her heart valve started to leak. She then contracted MRSA twice, was delayed in getting an intensive care bed and then died days after her operation.
Mrs Husband's family believe delays in transferring her from Hairmyres Hospital, in East Kilbride, to the Golden Jubilee, Clydebank, for cardiac surgery may have led to her death last month.
The two health boards have launched a investigation and Paul Gray, the chief executive of NHS Scotland, is now involved in the case.
Mrs Husband, a grandmother-of-two, who was from Strathaven, Lanarkshire had a heart valve replaced in January 2013 but was admitted to Hairmyres Hospital this year after doctors discovered the valve was leaking.
Doctors initially tried to stabilise her medically with drugs, but it became clear this was not working and her condition began to deteriorate.
However, it was more than two months before she was transferred to the Golden Jubilee for heart surgery.
Her family say a number of scheduled meetings that had been set up between doctors at the hospitals to discuss her condition did not take place.
They say the pensioner, who has two sons, Iain, who lives in Glasgow, and Stephen, who lives in Bath, also caught MRSA twice and another delay was caused after an ICU bed she was due to be transferred to was taken up by another patient.
By the time she was transferred for surgery, the family say she was seriously ill and she died last month shortly after the surgery.
Her brother Geoff Caine said the surgeon at the Golden Jubilee was "shocked" at her condition when she arrived at the hospital.
Mr Caine, who lives in Manchester, said: "On admission to Hairmyres, Marie appeared in good health, other than difficulty breathing caused by the heart valve leaking.
"The treatment involved administering the diuretic Furosemide, which carried a high risk of organ damage and could not be maintained for long periods at the high dose needed. It became obvious this wasn't the answer.
"There were several delays in the move to Golden Jubilee for the operation.
"There were missed meetings, she caught MRSA on two separate occasions and then, finally, the ICU bed she needed was booked for someone else.
"At this stage she began to lose hope and said she wanted to go to sleep and never wake up.
"We think she gave up the will to live, she could no longer stand the frustration of each promised transfer date arriving and passing without any positive action.
"In Marie's case something appears to have gone wrong with the communication between hospitals because the surgeon was shocked at her condition when she arrived for her operation.
"She was immediately put on life support and died after the subsequent operation.
"Her heart, kidneys, liver having failed, and her gut, plus a large part of her bowel, died due to lack of blood supply."
The family is calling for a review of the practice whereby patients can be kept at the Golden Jubilee for only one or two days prior to surgery.
They said Mrs Husband would have had a better chance of survival if she had been transferred sooner because of the specialised intensive care facilities for coronary patients.
Mr Caine, who has a brother Eric, said: "I think a wider discussion should take place to see if there are inherent problems with the way two hospitals share the care of a critically ill patient.
"Patients can only be kept at the Golden Jubilee Hospital for one or two days prior to the operation.
"We are sure this arrangement works for the majority of patients, but there are exceptions when they are in such a critical condition they need different arrangements.
"The ICU ward there has facilities that are far superior to those at Hairmyres Coronary Care Unit.
"We should stress we are not making any complaints against individual staff. We were moved by the compassion and dedication of staff of both hospitals.
"My sister was a bright articulate woman. She led a very full life and didn't look her age.
"There are lessons to be learnt from what happened and we hope improvements can be made to systems and procedures that reduce the chance of unnecessary deaths and suffering for critically ill patients."
A spokeswoman for the Golden Jubilee Hospital board said: "We are working alongside the family as well as our colleagues at NHS Lanarkshire to fully examine the issues raised.
"However, we need to await the outcome of this review to determine if there is any further action or improvements required. It is important we learn and continually improve and we are grateful to any patient, relative, friend or carer who takes the time to write and bring concerns to our attention."